Recursive differentiation method: application to the dynamics of beams on two parameter foundations

Abstract

The recursive differentiation method has been suggested and utilised to obtain analytical solutions for differential equations describe many boundary value problems. The method has been used to investigate an axially loaded beam resting on a two parameter foundation subjected to nonlinear lateral excitation. The obtained analytical solution expressions are similar to those obtained from other analytical techniques but with relatively less mathematical effort. Several examples are solved to describe the method and the obtained results reveal that the method is convenient for solving linear, nonlinear and higher order ordinary differential equations. It is found that in the case of beams on elastic foundations, the critical load corresponding to the first buckling is not always the smallest critical load. However, as the stiffness of the foundations increases relative to the beam, a critical load corresponding to one of the higher buckling modes may be smaller than the critical load corresponding to the first buckling mode which must be considered to avoid the buckling instability. Further, a similar phenomenon has been found for the natural frequencies of stressed beams on elastic foundations.

Keywords:

• boundary value problems
• recursive differentiation method
• differential equations
• natural frequencies
• critical loads
• beams on elastic foundation

Additional information

Notes on contributors

Mohamed Taha Hassan

Mohamed Taha Mohamed Hassan (Taha, M.H.) is an associate professor in the Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza, Egypt. He received his MSc and PhD in Engineering Mechanics in 1989 and 1995, respectively. Since 1982 he worked as a staff member in Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University. He also worked as a consultant engineer and general director of ALFACONSULT. His fields of interest include solid mechanics, structure dynamics, soil mechanics, analytical and numerical methods for solution of differential equations governing solid mechanics problems.