本文建構原子力顯微鏡(AFM)之微探針之動態測量數學模式。考慮具時變彈性基台，具阻尼與端點集中質量之微型樑，建立整體系統之統御偏微分運動方程和時變彈性邊界條件，以變數變換法將非齊次邊界條件轉換成齊次邊界，並提出一套解析法可有效的求解非線性邊界條件問題，此法並可廣泛應用於求解非線性邊界值問題。利用以上解法來研究非接觸量測模式(non-contact mode)之共振頻率偏移問題，當探針測量物件時，探針和表面間之非線性力作用，使得共振頻率有偏移之現象，比較在非量測時之探針共振頻率，得知共振頻率偏移量，進而推測得知精度可達 等級的表面微結構，最後，研究各種參數對暫態反應與共振頻率偏移之影響。

The study is to establish the dynamic measuring modes of microprobes of an atomic force microscopy. The forced vibration of a non-uniform beam with the time-dependent elastic boundary conditions, damping and concentrated tip mass is considered. The governing differential equation and the associated boundary conditions are derived by using the Hamilton’s principle. The non-homogeneous boundary conditions are transformed into homogeneous ones through the procedure of change of dependent variable. A new analytical solution for the system of non-contact mode is derived to solve the problem with nonlinear boundary condition. This method can be generally applied to solve the problems with nonlinear boundary conditions. By utilizing this method, the frequency shift of non-contact mode is investigated. When the microprobe approaches the sample surface, the nonlinear interaction between tip and sample causes a resonant frequency shift of the microprobe. Thus an atomic-scale surface image can be profiled with the frequency shift. Finally, the influence of the parameters on the transient response and the resonant frequency shift are investigated.

[1] Binnig, G., Roher, H., Gerber, H. and Weibel E. “Surface studied by scanning
tunneling microscopy,” Physical Review Letters, Vol. 49, pp. 57-61, 1982.

[2] Binnig, G., Quate, C. F., and Gerber, Ch., “Atomic force microscope,”
Physical Review Letters, Vol. 56, pp. 930-933, 1986.

[3] Holscher, H., Schwarz, U. D, Wiesendanger, R., “Calculation of the frequency
shift in dynamic force microscopy,” applied surface science, Vol.140,
pp. 344-351, 1999.

[4] Sasaki, N., Tsukada, M., Tamura,R., Abe,K.,Sato,N., “Dynamics of the
cantilever in noncontact atomic force microscopy,” Applied physics A
Materials, Vol.66, pp. 287-291, 1998.

[5] Rabe, U., Janser, K., and Amold,W., “Vibrations of free and surface-coupled
force microscope cantilevers : Theory and experiment,” American institute of
physics, Vol.67, pp. 3281-3293, 1996.

[6] Weigert, S., Dreier, M., and Hegner, M., “Fequency shifts of cantilevers
vibrating in various media,” Appl. phys. letter.,Vol.69, pp.2834-2836, 1996.

[7] Fung, R. F., Huang, S. C., “Dynamic modeling and vibration analysis of the
atomic force microscope,” transactions of ASME, Vol.123, pp. 502-509, 2001.

[8] Sokolov, I. Y., Henderson, G. S.,Wicks, F. J., “Model dependence of AFM
simulations in non-contact mode, ” surface science, Vol.457, pp. 267-272
, 2000.

[9] Kokubun, K., Murakami, H., Toda, Y. and Ono, M., “A bending and stretching
mode crystal oscillator as friction vacuum gauge,” Vacuum, Vol. 34, pp.
731-735, 1984.

[10] Blom, F.R., Bouwstra, S., Elwenspoek, M. and Fluitman, J.H.J., “Dependence
of the quality factor of micro-machined silicon beam resonators on pressure
and geometry,” J. vac. Sci. Technol., B10, pp. 19-26, 1992.

[11] Hirano, T., Funrehata, T., Gabriel, K. J. and Fujita, “Effect of air
damping and suspension compliance on electrostatic comb-drive actuators with
sub-micron gap,” Proc. 2nd Int. Symp. Measurement and Control in Robotics,
pp. 507-512, 1992.

[12] Murakami, Y. and Moronuki, N., “study on the design of micro-machine,” 1st
Report, Proc. Jpn. Soc. Precision Eng., pp. 919-920, 1993.

[13] Matsuda, R. and Fukui, S., “Asymptotic analysis of ultra-thin gas squeeze
film lubrication for infinitely large squeeze number,” ASME Paper
94-Trib-2, 1994.

[14] Darling, R. B., Hivick, C., and Xu, J., “Compact analytical modeling
squeeze film damping with arbibratrary venting conditions using a Green’s
function approach,” Sensors and Actuators, Vol. A70, pp. 32-41, 1998.

[15] Xu, Y. and Smith, S. T, “Determination of squeeze film damping in
capacitance-based cantilever force probes,” Precision Engineering, Vol. 17,
pp. 94-100, 1995.

[16] Lee, S. Y. and Lin, S. M., “Dynamic analysis of non-uniform beams with time
dependent elastic boundary conditions,” ASME Journal of Applied Mechanics,
Vol. 63, pp. 474-478, 1996.

[17] Lee, S. Y. and Kuo,Y. H., “Exact solutions for the analysis of general
elastically restrained nonuniform beams,” ASME Journal of Applied
Mechanics, Vol. 59, No. 2, pp. 205-212, 1992.

[18] Israelachvili, J. N., “Intermolecular and Surface Forces,” Academic Press,
New York, 1985.

[19] Sarid, D., “Scanning Force Microscopy With Application to Electric,Magnetic
and Atomic Forces,” Oxford University Press, New York, 1994.

[20] Meirovitch, L., “Fundamentals of Vibrations,” McGraw-Hill, New York, 2001.

[21] Rao, J. S., “Advanced Theory of Vibration,” Wiley Eastern Limited, 1992.

[22] Hosaka, H., Itao K., Kuroda S., “Damping Characteristics of Beam-Shaped
Micro-oscillators,” Sensors and Actuators, Vol. A70, pp. 32-41, 1998.

[23] Lee, S. Y., Wang, W. R. and Chen, T. Y., “A General Approach on the
Mechanical Analysis of Nonuniform Beams With Nonhomogeneous Elastic
Boundary Conditions,” J. Vibration and Acoustics, A49, pp. 87-95, 1995.

[24] Guggisberg, M., Bammerlin, M., Loppacher, C., Pfeiffer, O., Abdurrixit, A.,
Barwich, V., Bennewitz, R., Baratoff, A., Meyer, E., “Separation of
interaction by non-contact force microscopy,” Physical Review B, Vol. 61,
pp.11151-11155, 2000