摘要欧拉-伯努利梁模型和一维弹簧振子模型是原子力显微镜（Atomic Force Microscopy，AFM）的动力学理论基础。文章提出了计算等效阻尼的新方法，并分别用挠曲线函数和一阶振型函数两种简化方式计算了由欧拉-伯努利梁模型简化为一维弹簧振子模型的等效质量、等效刚度和等效阻尼，并对两种简化方式进行了比较。在各项参数等效的基础上，进一步对固定端位移激励下悬臂梁的响应振幅进行了探讨。最后设计了无针尖探针远离样品处的扫频实验，实验表明用一阶振型函数简化所得的理论结果和实验吻合较好。该结果为原子力显微镜的动力学特性研究提供了理论参考。
Abstract：Euler-Bernoulli beam model and one-dimensional spring oscillator model are the dynamic theoretical basis of atomic force microscope (AFM). In this paper, a new method for calculating the equivalent damping was proposed, and the equivalent mass, equivalent stiffness and equivalent damping simplified from Euler-Bernoulli beam model to one-dimensional spring vibrator model were calculated using two simplified methods: deflection curve function and first-order vibration mode function, and the two simplified methods were compared. On the basis of equivalent parameters, the response amplitude of cantilever beam excited with fixed end displacement was further discussed. Finally, the frequency sweep experiment of the needle free probe away from the sample was designed. The experimental results show that the theoretical results obtained by simplifying the first-order mode function are in good agreement with the experiment. The results provide a theoretical reference for the study of the kinetic properties of atomic force microscope.
宾凤姣，关睿，刘国林，曾瑜，魏征. 原子力显微镜动力学简化模型的振动参数探讨[J]. 振动与冲击, 2023, 42(6): 240-247.
BIN Fengjiao, GUAN Rui, LIU Guolin, ZENG Yu, WEI Zheng. Discussion on vibration parameters of the simplified dynamic model of an atomic force microscope. JOURNAL OF VIBRATION AND SHOCK, 2023, 42(6): 240-247.
[1] GARCIA R. Amplitude modulation atomic force microscopy [M]. John Wiley & Sons, 2011.
[2] BINNIG G, QUATE C F, GERBER. Atomic force microscope [J]. Physical Review Letters, 1986, 56: 930–933.
[3] GARCIA R, PEREZ R. Dynamic atomic force microscopy methods [J]. Surface Science Reports, 2002, 47(6-8): 197-301.
[4] 白春礼, 田芳, 罗克. 扫描力显微术[M]. 北京: 科学出版社, 2000, 1-4.
BAI Chunli, TIAN Fang, LUO Ke. Scanning force microscopy [M]. Science Press, 2000, 1-4.
[5] JALILI N, LAXMINARAYANA K. A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences [J]. Mechatronics, 2004, 14(8):907-945.
[6] GIESSIBL F J. Advances in atomic force microscopy [J]. Reviews of Modern Physics, 2003, 75(3) : 949-983.
[7] BINNIG G, ROHRER H. In touch with atoms [J]. Review of Modern Physics, 1999, 71(2) :324-330.
[8] MELCHER J, HU S, RAMAN A. Equivalent point-mass models of continuous atomic force microscope probes [J]. Applied Physics Letters, 2007, 91(5): 053101.
[9] STARK M, STARK R W, HECKL W M, et al. Inverting dynamic force microscopy: From signals to time-resolved interaction forces [J]. Proceedings of the National Academy of Sciences, 2002, 99(13): 8473-8478.
[10] BUTT H J, JASCHKE M. Calculation of thermal noise in atomic force microscopy [J]. Nanotechnology, 1995, 6(1): 1.
[11] LEE S I, HOWELL S W, RAMAN A, et al. Nonlinear dynamics of microcantilevers in tapping mode atomic force microscopy: A comparison between theory and experiment [J]. Physical Review B, 2002, 66(11): 115409.
[12] SADER J E, LARSON I, MULVANEY P, et al. Method for the calibration of atomic force microscope cantilevers [J]. Review of Scientific Instruments, 1995, 66(7): 3789-3798.
[13] CLIFFORD C A, SEAH M P. The determination of atomic force microscope cantilever spring constants via dimensional methods for nanomechanical analysis [J]. Nanotechnology, 2005, 16(9): 1666.
[14] RAMAN A, MELCHER J, TUNG R. Cantilever dynamics in atomic force microscopy[J]. Nano Today, 2008, 3(1-2): 20-27.
[15] SALAPAKA M V, BERGH H S, LAI J, et al. Multi-mode noise analysis of cantilevers for scanning probe microscopy [J]. Journal of Applied Physics, 1997, 81(6): 2480-2487.
[16] SAN P, GARCIA R. Tip-surface forces, amplitude, and energy dissipation in amplitude-modulation (tapping mode) force microscopy [J]. Physical Review B, 2001, 64(19): 193411.
[17] WINKER R G, SPATZ J P, SHEIKO S, et al. Imaging material properties by resonant tapping-force microscopy: A model investigation [J]. Physical Review B, 1996, 54(12): 8908.
[18] ANCZYKOWSKI B , GOTSMANN B , FUCHS H, et al. How to measure energy dissipation in dynamic mode atomic force microscopy [J]. Applied Surface Science, 1999, 140(3-4):376-382.
[19] SEBASTIAN A, SALAPAKA M V, CHEN D J, et al. Harmonic and power balance tools for tapping-mode atomic force microscope [J]. Journal of Applied Physics, 2001, 89(11):6473-6480.
[20] HOELSCHER H. Quantitative measurement of tip-sample interactions in amplitude modulation atomic force microscopy [J]. Applied Physics Letters, 2006, 89(12):4723.
[21] JESSE S, KALININ S V, PROKSCH R, et al. The band excitation method in scanning probe microscopy for rapid mapping of energy dissipation on the nanoscale [J]. Nanotechnology, 2007, 18(43):733-737.
[22] CLEVELAND J P, ANCZYKOWSKI B, SCHMID A E, et al. Energy dissipation in tapping-mode atomic force microscopy [J]. Applied Physics Letters, 1998, 72(20):2613-2615.
[23] SADER J E. What is the oscillation amplitude of a vibrating cantilever? [J]. Review of Scientific Instruments, 2019, 90(8): 086103.
[24] 刘延柱, 陈文良, 陈立群. 振动力学, 第2版[M]. 高等教育出版社, 2011.
LIU Yanzhu, CHEN Wenliang, CHEN Liqun. Vibration mechanics, Version 2 [M]. Higher Education Press, 2011.
[25] PAYAM A F, FATHIPOUR M. Effect of tip mass on frequency response and sensitivity of AFM cantilever in liquid [J]. Micron, 2014, 70C:50-54.
[26] GARCIA R, PEREZ R. Dynamic atomic force microscopy methods [J]. Surface Science Reports, 2002, 47(6-8):197-301.
[27] GOTSMANN B, FUCHS H. Dynamic AFM using the FM technique with constant excitation amplitude [J]. Applied Surface Science, 2002, 188(3-4):355-362.
[28] PAYTON O D, PICCO L, MILES M J, et al. Modelling oscillatory flexure modes of an atomic force microscope cantilever in contact mode whilst imaging at high speed [J]. Nanotechnology, 2012, 23(26): 265702.
[29] 刘鸿文，林建兴，曹曼玲.材料力学，第5版[M].高等教育出版社，2011.
LIU Hongwen, LIN Jianxing, CAO Manling. Mechanics of Materials, Version 5 [M]. Higher Education Press, 2011.
[30] MILLER J, ANSARI A, HEINZ D B, et al. Effective quality factor tuning mechanisms in micromechanical resonators[J]. Applied Physics Reviews, 2018, 5(4):041307.
[31] 魏征, 郑骁挺, 刘晶等. 轻敲模式下AFM动力学模型及能量耗散机理研究 [J].力学学报, 2020,52(04):1106-1119.
WEI Zheng, ZHENG Xiaoting, LIU Jing, et al. Research on AFM dynamic model and energy dissipation mechanism in tapping mode [J]. Acta Mechanica Sinica, 2020,52(04):1106-1119.
[32] WEI Z, LIU J, ZHENG X T, et al. Influence of squeeze film damping on quality factor in tapping mode atomic force microscope [J]. Journal of Sound and Vibration, 2020, 491(23):115720.
[33] 柳世华, 魏征, 孙岩等. 压膜阻尼对原子力显微镜振动特性的影响研究 [J]. 振动与冲击, 2020, 39(11): 185-191.
LIU Shihua, WEI Zheng, SUN Yan, et al. Effect of film damping on vibration characteristics of atomic force microscope [J]. Journal of Vibration and Shock, 2020, 39(11): 185-191.
[34] HIROSHI, HOSAKA, KIYOSHI, et al. Damping characteristics of beam-shaped micro-oscillators [J]. Sensors and Actuators A: Physical, 1995, 49(1-2):87-95.
[35] CHEN G Y, WARMACK R J, THUNDAT T, et al. Resonance response of scanning force microscopy cantilevers [J]. Review of Scientific Instruments, 1994, 65(8):2532-2537.
[36] LANDAU L D, LIFSHITZ E M. Fluid Mechanics:Course of Theoretical Physics. Pergamon Press, 2013.
