Green functions for four-point boundary value problems with applications to heterogeneous beams

Abstract

The main objective of this study is to define the Green functions for four-point boundary value problems. It is a further aim to clarify what properties the Green functions have and to present a method for calculating the elements of these Green functions. The examples are related to two heterogeneous beams with four supports: the (first) [second] beam is (fixed)[pinned] at the endpoints while the intermediate supports are two rollers. Determination of the eigenfrequencies leads to four-point eigenvalue problems associated with homogeneous boundary conditions. Utilizing the Green functions that belong to these eigenvalue problems we can transform those into eigenvalue problems governed by homogeneous Fredholm integral equations. Then a numerical solution is computed by reducing the homogeneous Fredholm integral equations into algebraic eigenvalue problems.

Keywords

• Beam
• Four-point boundary value problem
• Green function
• Heterogeneous
• Eigenvalue problem
• Vibrations