本論文主要在討論反算法於尤拉樑振動問題之研究。在實際工程上，作用於樑上的外力或力矩可能無法直接由量測或是計算求得，這些物理量將可由反算法中的共軛梯度法(Conjugate Gradient Method)配合位移量之量測值來預測之。本論文可分為兩個章節，第一章為共軛梯度法於切削工具所承受未知外力之預測，第二章為共軛梯度法於懸臂樑末端未知時變外力及力矩之同時預測。
第一章旨在利用反算法中的共軛梯度法來進行反算分析工作。本問題的物理模型為一端夾住，另一端自由之切削刀具，所要預測的物理量為作用於刀具前端切削區域的未知外力，此外力為位置及時間的函數。本文將以量測所得之位移資料為依據來修正所預測之未知外力。在本章之探討中，將分別考慮在不同量測誤差，不同量測區域下且當外力的初始猜值為零時之預測結果來驗證此反算法的正確性。
第二章亦是利用反算法中的共軛梯度法來進行反算分析工作。本問題的物理模型為一端夾住且另一端自由之懸臂樑，所要預測的物理量為作用於懸臂樑末端未知的時變外力及力矩，並將利用量測之位移資料為依據來修正所預測之外力及力矩。本章中將分別考慮在不同量測誤差，不同量測位置下且當外力及力矩的初始猜值為零時之預測結果以驗證此反算法的正確性。

In practical engineering problem, there exist many physical quantities that are very difficult to measure directly. The techniques for“INVERSE PROBLEM”can be used to solve these kinds of problems. In the present thesis the inverse vibration problem for an Euler-Bernoulli beam is discussed.
An inverse forced vibration problem, based on the Conjugate Gradient Method (CGM), (or the iterative regularization method), is examined in chapter 1 to estimate the unknown spatial and temporal-dependent external forces for the cutting tools,and in chapter 2 to estimate simultaneously the unknown time-dependent applied force and moment for an Euler-Bernoulli beam by utilizing the simulated beam displacement measurements. The tool is represented by an Euler-Bernoulli beam. The accuracy of the inverse analysis is examined by using the simulated exact and inexact displacement measurements. The numerical experiments are performed to test the validity of the present algorithm by using different types of external forces, sensor arrangements and measurement errors. Results show that excellent estimations on the external forces can be obtained with any arbitrary initial guesses.

1. L. Andren, L. Hakansson A. Brandt and I. Claesson, 2004 Mechanical Systems and Signal Processing 18, 903–927. Identification of Motion of Cutting Tool Vibration in a Continuous Boring Operation—Correlation to Structural Properties.
2. G.M.L. Gladwell 1986 Inverse Problem in Vibration. The Netherlands: Kluwer Academic Publishers.
3. L. Starek and D. J. Inman 1991 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 58, 1101-1104. On the inverse vibration problem with rigid-body modes.
4. L. Starek, D. J. Inman and A. Kress 1992 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics 114, 565-568. A symmetric inverse vibration problem.
5. L. Starek and D. J. Inman 1995 Journal of sound and Vibration 181, 893-903. A symmetric inverse vibration problem with overdamped modes.
6. G. Desanghere and R. Snoeys 1985 Proceedings of the 1985 International Modal Analysis Conference, Orlando, FL, 685-690. Indirect identification of excitation force by modal coordinate transformation.
7. K. K. Stevens 1987 Proceedings of the SEM Spring Conference on Experimental Mechanics, Houston, TX, June 14-19, 838-844, Force identification problems - an overview,.
8. V. I. Bateman, T. G. Carne, D. L. Gregory, S. W. Attaway and H. R. Yoshimura 1991 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics 113, 192-200. Force reconstruction for impact tests.
9. J. E Michaels and Y. H. Pao 1985 Journal of Acoustical Society of American, 77, 2005-2011. The inverse source problem for an oblique force on an elastic plate.
10. C. K. Ma, P. C. Tuan, D. C. Lin and C. S. Liu 1998 International Journal of System Science, 29, 663-672. A Study of an Inverse Method for Impulsive Loads Estimation.
11. C. H. Huang 2001 J. Sound and Vibration, 242, 749-756. An Inverse Nonlinear Force Vibration Problem of Estimating the External Forces in a Damped System with Time-Dependent System Parameters.
12. C. H. Huang 2005 Applied Mathematical Modelling, 29, 1022–1039. A Generalized Inverse Force Vibration Problem for Simultaneously Estimating the Time-Dependent External Forces.
13. T.H.T. Chan and D. B. Ashebo 2006 Journal of Sound and Vibration, 295, 870–883. Theoretical Study of Moving Force Identification on Continuous Bridges.
14. H. L. Lee 2006 Ultramicroscopy, 106, 547–552. Inverse Estimation of the Tapered Probe-Sample Shear Force of Scanning Near-Field Optical Microscope.
15. C. H. Huang, C. Y. Yeh and Helcio R. B. Orlande 2004 Chemical Engineering Science, 58, 3741-3752. A Non-Linear Inverse Problem in Simultaneously Estimating the Heat and Mass Production Rates for A Chemically Reacting Fluid.
16. C. H. Huang and C. Y. Huang 2004 Int. J. Heat and Mass Transfer, 47, 447-457. An Inverse Biotechnology Problem in Estimating the Optical Diffusion and Absorption Coefficients of Tissue.
17. C. H. Huang and H. C. Lo 2005 Numerical Heat Transfer, part A-Applications, 48, 1009-1034. A Three-Dimensional Inverse Problem in Predicting the Heat Fluxes Distribution in the Cutting Tools.
18. C. H. Huang and J. X. Li 2006 Journal of Physics D: Applied Physics, 39, 2343–2351. A Non-Linear Optimal Control Problem in Obtaining Homogeneous Concentration for Semiconductor Materials.
19. C. H. Huang and H. C. Lo 2006 Applied Thermal Engineering, 26, 1515–1529. A Three-Dimensional Inverse Problem in Estimating the Internal Heat Flux of Housing for High Speed Motors.
20. O. M. Alifanov 1994 Inverse Heat Transfer Problems, Springer-Verlag, Berlin Heidelberg.
21 L. S. Lasdon, S. K. Mitter and A. D. Warren 1967 IEEE Transactions on Automatic Control. AC-12, 132-138. The Conjugate Gradient Method for Optimal Control Problem.
22. IMSL Library Edition 10.0, 1987 IMSL, Houston, TX. User's Manual: Math Library Version 1.

1. G.M.L. Gladwell 1986 Inverse Problem in Vibration. The Netherlands: Kluwer Academic Publishers.
2. G. Desanghere and R. Snoeys 1985 Proceedings of the 1985 International Modal Analysis Conference, Orlando, FL, 685-690. Indirect identification of excitation force by modal coordinate transformation.
3. K. K. Stevens 1987 Proceedings of the SEM Spring Conference on Experimental Mechanics, Houston, TX, June 14-19, 838-844, Force identification problems - an overview.
4. L. Starek and D. J. Inman 1991 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 58, 1101-1104. On the inverse vibration problem with rigid-body modes.
5. L. Starek, D. J. Inman and A. Kress 1992 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics 114, 565-568. A symmetric inverse vibration problem.
6. L. Starek and D. J. Inman 1995 Journal of sound and Vibration 181, 893-903. A symmetric inverse vibration problem with overdamped modes.
7. V. I. Bateman, T. G. Carne, D. L. Gregory, S. W. Attaway and H. R. Yoshimura 1991 Transactions of the American Society of Mechanical Engineers, Journal of Vibration and Acoustics 113, 192-200. Force reconstruction for impact tests.
8. J. E Michaels and Y. H. Pao 1985 Journal of Acoustical Society of American, 77, 2005-2011. The inverse source problem for an oblique force on an elastic plate.
9. C. K. Ma, P. C. Tuan, D. C. Lin and C. S. Liu 1998 International Journal of System Science, 29, 663-672. A Study of an Inverse Method for Impulsive Loads Estimation.
10. C. H. Huang 2001 J. Sound and Vibration, 242, 749-756. An Inverse Nonlinear Force Vibration Problem of Estimating the External Forces in a Damped System with Time-Dependent System Parameters.
11. C. H. Huang 2005 Applied Mathematical Modelling, 29, 1022–1039. A Generalized Inverse Force Vibration Problem for Simultaneously Estimating the Time-Dependent External Forces.
12. T.H.T. Chan and D. B. Ashebo 2006 Journal of Sound and Vibration, 295, 870–883. Theoretical Study of Moving Force Identification on Continuous Bridges.
13. H. L. Lee 2006 Ultramicroscopy, 106, 547–552. Inverse Estimation of the Tapered Probe-Sample Shear Force of Scanning Near-Field Optical Microscope.
14. J. D. Chang and B. Z. Guo, 2007, Automatica, 43, 732-737. Identification of Variable Spatial Coefficients for a Beam Equation from Boundary Measurements.
15. C. H. Huang and J. X. Li 2006 Journal of Physics D: Applied Physics, 39, 2343–2351. A Non-Linear Optimal Control Problem in Obtaining Homogeneous Concentration for Semiconductor Materials.
16. C. H. Huang and H. C. Lo 2006 Applied Thermal Engineering, 26, 1515–1529. A Three-Dimensional Inverse Problem in Estimating the Internal Heat Flux of Housing for High Speed Motors.
17. C. H. Huang, L. C. Jan, R. Li and A. J. Shih 2007 Int. J. Heat and Mass Transfer, Vol. 50, No. 17-18, pp. 3265-3277. A Three-Dimensional Inverse Problem in Estimating the Applied Heat Flux of a Titanium Drilling -- Theoretical and Experimental Studies.
18. C. H. Huang and H. H. Wu 2006 Int. J. Heat and Mass Transfer, Vol. 49, No. 25-26, pp. 4893–4902. An Iterative Regularization Method in Estimating the Base Temperature for Non-Fourier Fins.
19. C. H. Huang and K.Y. Chen 2007 Int. J. Numerical Methods in Engineering, Vol. 69, No. 7, pp. 1499–1520. An Inverse Phonon Radiative Transport Problem in Estimating the Boundary Temperatures for Nanoscale Thin Films.
20. O. M. Alifanov 1994 Inverse Heat Transfer Problems, Springer-Verlag, Berlin Heidelberg.
21 L. S. Lasdon, S. K. Mitter and A. D. Warren 1967 IEEE Transactions on Automatic Control. AC-12, 132-138. The Conjugate Gradient Method for Optimal Control Problem.
22. IMSL Library Edition 10.0, 1987 IMSL, Houston, TX. User's Manual: Math Library Version 1.