Green’s function for uniform Euler-Bernoulli beams at resonant condition: introduction of Fredholm alternative theorem. (English) Zbl 1443.74210
Summary: This paper deals with the dynamic analysis of Euler-Bernoulli beams at the resonant condition. The governing partial differential equation of the problem is converted into an ordinary differential equation by applying the well-known Fourier transform. The solution develops a Green’s function method which involves establishing the Green’s function of the problem, applying the pertinent boundary conditions of the beam. Due to the special conditions of the resonant situation, a significant obstacle arises during the derivation of the Green’s function. In order to overcome this hurdle, however, the Fredholm Alternative Theorem is employed; and it is shown that the modified Green’s function of the beam may still be achievable. Furthermore, the necessary requirement so that the resonant response will be found is introduced. A special case which refers to a case in the absence of resonance is also included, for some verification purposes.
MSC:
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 35A08 | Fundamental solutions to PDEs |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
References:
| [1] | Kukla, S., The Green’s function method in frequency analysis a beam with intermediate elastic supports, J. Sound Vibr., 149, 154-159 (1991) |
| [2] | Mohamad, A. S., Tables of Green’s function for the theory of beam vibrations with general intermediate appendages, Int. J. Solids Struct., 31, 93-102 (1994) · Zbl 0800.73184 |
| [3] | Lueschen, G. G.G.; Bergman, L. A., Green’s functions for uniform Timoshenko beams, J. Sound Vibr., 194, 93-102 (1996) |
| [4] | Foda, M. A.; Abduljabbar, Z., A dynamic Green function formulation for the response of a beam structure to a moving mass, J. Sound Vibr., 210, 295-306 (1998) |
| [5] | Abu-Hilal, M., Forced vibration of Euler-Bernoulli beams by means of dynamic Green function, J. Sound Vibr., 267, 191-207 (2003) · Zbl 1236.74136 |
| [6] | Kukla, S.; Zamojska, I., Frequency analysis of axially loaded stepped beams by Green’s function method, J. Sound Vibr., 300, 1034-1041 (2007) |
| [7] | Mehri, B.; Davar, A.; Rahmani, O., Dynamic Green function solution of beams under a moving load with different boundary conditions, Sci. Iran., 16, 273-279 (2009) |
| [8] | Failla, G., Closed-form solutions for Euler-Bernoulli arbitrary discontinuous beams, Arch. Appl. Mech., 81, 605-628 (2011) · Zbl 1271.74260 |
| [9] | Qin, Q. H., Green’s Function and Boundary Elements of Multifield Materials (2007), Elsevier: Elsevier U.K. |
| [10] | King, A. C.; Billingham, J.; Otto, S. R., Differential Equations (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1034.34001 |
| [11] | Korn, G. A.; Korn, T. M., Mathematical Handbook for Scientists and Engineers (1968), McGraw-Hill: McGraw-Hill U.S. · Zbl 0177.29301 |
| [12] | Leissa, A. W.; Qatu, M. S., Vibrations of Continuous Systems (2011), McGraw-Hill: McGraw-Hill U.S. |
| [14] | Dominguez, J., Boundary Elements in Dynamics (1993), WIT Press: WIT Press Southampton · Zbl 0790.73003 |
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