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Analyzing Free-Free Beams by Green's Functions and Fredholm Alternative Theorem

Published online by Cambridge University Press:  10 July 2017

M. Rezaiee-Pajand ,
A. Aftabi Sani  and
S. M. Hozhabrossadati
M. Rezaiee-Pajand*
Affiliation:
Department of Civil EngineeringFerdowsi University of MashhadMashhad, Iran
A. Aftabi Sani
Affiliation:
Department of Civil EngineeringFerdowsi University of MashhadMashhad, Iran
S. M. Hozhabrossadati
Affiliation:
Department of Civil EngineeringFerdowsi University of MashhadMashhad, Iran
*
*Corresponding author ([email protected])
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Abstract

This article deals with the analysis of free-free beams by an analytical method. The well-known Green's function method is employed, and exact solution for the problem is obtained. As a second problem, the simply supported-free beam with rotational rigid body motion is analyzed. It is initially shown that ordinary Green's functions cannot be constructed due to a mathematical contradiction. To remedy this limitation, the Fredholm Alternative Theorem is utilized. This theorem eliminates the contradiction and enables analysts to obtain modified Green's functions. The fundamental existence conditions are derived and thoroughly investigated from the structural point of view. Finally, the deflection functions of these beams are found using modified Green's functions.

Keywords

Modified Green's functionsFredholm Alternative TheoremDeflection functionsSolvability conditions
Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 1 , February 2019 , pp. 27 - 39
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2019 

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References

1. Bhagat, M. and Ganguli, R., “Spatial Fourier Analysis of a Free-Free Beam for Structural Damage Detection,” International Journal for Computational Methods in Engineering Science and Mechanics, 15, pp. 356–371 (2014).Google Scholar
2. Sarkar, K. and Ganguli, R., “Closed-Form Solutions for Non-Uniform Euler-Bernoulli Free-Free Beams,” Journal of Sound and Vibration, 332, pp. 6078–6092 (2013).Google Scholar
3. Alonso, J. S. and Burdisso, R. S., “Green's Functions for the Acoustic Field in Lined Ducts with Uniform Flow,” AIAA Journal, 45, pp. 2677–2687 (2007).Google Scholar
4. Caddemi, S., Calio, I. and Cannizzaro, F., “Closed-Form Solutions for Stepped Timoshenko Beams with Internal Singularities and Along-Axis External Supports,” Archive of Applied Mechanics, 83, pp. 559–577 (2013).Google Scholar
5. Chen, J. T., Liao, H. Z. and Lee, W. M., “An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains,” Journal of Mechanics, 25, pp. 59–74 (2009).Google Scholar
6. Eskandary, M. and Ahmadi, F., “Green's Functions of a Surface-Stiffened Transversely Isotropic Half-Space,” International Journal of Solids and Structures, 49, pp. 3282–3290 (2012).Google Scholar
7. Failla, G. and Santini, A., “On Euler-Bernoulli Discontinuous Beam Solutions via Uniform-Beam Green's Functions,” Mechanics Research Communications, 44, pp. 7666–7687 (2007).Google Scholar
8. Kelkel, K., “Green's Function and Receptance for Structures Consisting of Beams and Plates,” AIAA Journal, 25, pp. 1482–1489 (1987).Google Scholar
9. Linton, C. M., “A New Representation for the Free-Surface Channel Green's Function,” Applied Ocean Research, 21, pp. 17–25 (1999).Google Scholar
10. Melnikov, Y. A., “Influence Functions of a Point Force for Kirchhoff Plates with Rigid Inclusions,” Journal of Mechanics, 20, pp. 249–256 (2004).Google Scholar
11. Taylor, R. E. and Ohkusu, M., “Green Functions for Hydroelastic Analysis of Vibrating Free-Free Beams and Plates,” Applied Ocean Research, 22, pp. 295–314 (2000).Google Scholar
12. Nicholson, J. W. and Bergman, L. A., “Free Vibration of Combined Dynamical Systems,” Journal of Engineering Mechanics, 112, pp. 1–13 (1986).Google Scholar
13. Mohamad, A. S., “Tables of Green's Functions for the Theory of Beam Vibrations with General Intermediate Appendages,” International Journal of Solids and Structures, 31, pp. 257–268 (1994).Google Scholar
14. Kukla, S., “Application of Green Functions in Frequency Analysis of Timoshenko Beams with Oscillators,” Journal of Sound and Vibration, 205, pp. 355–263 (1997).Google Scholar
15. Foda, M. A. and Abduljabbar, Z., “A Dynamic Green Function Formulation for the Response of a Beam Structure to a Moving Mass,” Journal of Sound and Vibration, 210, pp. 295–306 (1998).Google Scholar
16. Abu-Hilal, M., “Forced Vibration of Euler-Bernoulli Beams by Means of Dynamic Green's Functions,” Journal of Sound and Vibration, 267, pp. 191–207 (2003).Google Scholar
17. Kukla, S. and Zamojska, I., “Frequency Analysis of Axially Loaded Stepped Beams by Green's Function Method,” Journal of Sound and Vibration, 300, pp. 1034–1041 (2007).Google Scholar
18. Mehri, B., Davar, A. and Rahmani, O., “Dynamic Green Function Solution of Beams under a Moving Load with Different Boundary Conditions,” Scientia Iranica, 16, pp. 273–279 (2009).Google Scholar
19. Failla, G., “Closed-Form Solutions for Euler-Bernoulli Arbitrary Discontinuous Beams,” Archive of Applied Mechanics, 81, pp. 605–628 (2011).Google Scholar
20. Ghannadiasl, A. and Mofid, M., “Dynamic Green Function for Response of Timoshenko Beam with Arbitrary Boundary Conditions,” Mechanics Based Design of Structures and Machines, 42, pp. 97–110 (2014).Google Scholar
21. Li, X. Y., Zhao, X. and Li, Y. H., “Green's Functions of the Forced Vibration of Timoshenko Beams with Damping Effect,” Journal of Sound and Vibration, 333, pp. 1781–1795 (2014).Google Scholar
22. Hozhabrossadati, S. M. and Aftabi Sani, A., “Application of Green's Function for Constructing Influence Lines,” Journal of Engineering Mechanics, 142, 04015097.Google Scholar
23. Akkaya, T. and Horssen, W. T., “On Constructing a Green's Function for Semi-Infinite Beam with Boundary Damping,” Meccanica, DOI:10.1007/s11012-016-0594-9 (2016).Google Scholar
24. Hozhabrossadati, S. M., Aftabi Sani, A., Mehri, B. and Mofid, M., “Green's Function for Uniform Euler-Bernoulli Beams at Resonant Condition: Introduction of Fredholm Alternative Theorem,” Applied Mathematical Mode lling, 39, pp. 3366–3379 (2015).Google Scholar
25. Hozhabrossadati, S. M. and Aftabi Sani, A., “Deformation of Euler-Bernoulli Beams by Means of Modified Green's Function: Application of Fredholm Alternative Theorem,” Mechanics Based Design of Structures and Machines, 43, pp. 277–293 (2015).Google Scholar
26. Chen, J. T., Huang, W. S., Lee, J. W. and Tu, Y. C., “A Self-Regularized Approach for Deriving the Free-Free Flexibility and Stiffness Matrices,” Computers & Structures, 145, pp. 12–22 (2014).Google Scholar
27. Chen, J. T., Chang, Y. L., Leu, S. Y. and Lee, J. W., “Static Analysis of the Free-Free Trusses by Using a Self-Regularization Approach,” Journal of Mechanics, Available Online, DOI: 10.1017/jmech.2017.15 (2017).Google Scholar
28. Chen, J. T., Han, H., Kuo, S. R. and Kao, S. K., “Regularization Methods for Ill-Conditioned System of the Integral Equation of the First Kind with the Logarithmic Kernel,” Inverse Problems in Science and Engineering, 22, pp. 1176–1195 (2014).Google Scholar
29. Martin, P. A., “Multiple Scattering and Modified Green's Functions,” Journal of Mathematical Analysis and Applications, 275, pp. 642–656 (2002).Google Scholar
30. Greenberg, M., Application of Green's Functions in Science and Engineering, Prentice Hall, New Jersey (1971).Google Scholar
31. Korn, G. A. and Korn, T. M., Mathematical Handbook for Scientists and Engineers, Second Edition, McGraw-Hill, New York (1968).Google Scholar
32. Stakgold, I., Green's Functions and Boundary Value Problems, John Wiley & Sons, New York (1979).Google Scholar
33. Hibbeler, R. C., Structural Analysis, Sixth Edition, Prentice Hall, Singapore (2006).Google Scholar