Flutter Analysis of Slender Beams with Variable Cross Sections Based on Integral Equation Formulation

Authors: Z. El Felsoufi, L. Azrar

Abstract:

This paper studies a mathematical model based on the integral equations for dynamic analyzes numerical investigations of a non-uniform or multi-material composite beam. The beam is subjected to a sub-tangential follower force and elastic foundation. The boundary conditions are represented by generalized parameterized fixations by the linear and rotary springs. A mathematical formula based on Euler-Bernoulli beam theory is presented for beams with variable cross-sections. The non-uniform section introduces non-uniformity in the rigidity and inertia of beams and consequently, more complicated equilibrium who governs the equation. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebro-differential system related to internal and boundary unknowns. A generalized formula for the deflection, the slope, the moment and the shear force are presented. The free vibration of non-uniform loaded beams is formulated in a compact matrix form and all needed matrices are explicitly given. The dynamic stability analysis of slender beam is illustrated numerically based on the coalescence criterion. A realistic case related to an industrial chimney is investigated.

Keywords: Chimney, BEM and integral equation formulation, non uniform cross section, vibration and Flutter.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1109305

References:

[1] Rosa M. A. Auciello N. M. Free vibrations of tapered beams with flexible ends. Journal of Computers and Structures 1996; 60 N°2: 197- 202.
[2] Naguleswaran S. Comments on vibration of non-uniform rods and beams. Journal of Sound and Vibration 1996; 195: 197-202.
[3] Wu J. S., Hsieh M. Free vibration analysis of a non uniform beam with multiple point masses. Structural Engineering and Mechanics 2000; 9: 449-467.
[4] Chen D. W., Wu J-S. The exact solution for the natural frequencies and mode shapes of non uniform beam with multiple spring-mass systems. Journal of Sound and Vibration 2002; 255 (2): 299-232.
[5] El Felsoufi Z., Azrar L. Buckling, flutter and vibration analyses of beams by integral equation formulations. Computers and Structures 2005; 83: 2632-2649
[6] El Felsoufi Z., Azrar L. Integral equation formulation and analysis of the dynamic stability of damped beams subjected to sub-tangential follower forces. Journal of Sound and Vibration; 296 (2006) 690–713
[7] Nardini D., Brebbia CA. A new approach to free vibration analysis using boundary elements. In: Brebbia CA, editor. Boundary element method in engineering, Berlin: Springer, 1982: 313-26.
[8] Beskos D. E. Boundary element methods in dynamic analysis. Part II (1986-1996). Appl. Mech. Rev. ASME 1997; 50: 149-97.
[9] Karimi Khorramadi and A.R. Nezamabadi. Vibration Analysis of Functionally Graded Engesser- Timoshenko Beams Subjected to Axial Load Located on a Continuous Elastic Foundation, World Academy Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial and Mechatronics Engineering Vol:8, No:10, 2015.
[10] Timoshenko SV, Gere JM. Theory of Elastic Stability. New York: McGraw-Hill, 1961.
[11] El Felsoufi Z., Hachi H. Conception et Réalisation d’un dispositif de mesure de vibration de l’OCP, Maroc Phosphore I; PFE, EMI, Rabat, Maroc, 1994.