Solutions for the vibration and stability problems of heterogenous beams with three supports using Green functions

The goal of this study is to calculate the eigenvalues that provide the eigenfre- quencies and the critical loads for two heterogeneous beams with three supports: the (first) [second] beam is (fixed)[pinned] at the left end, the intermediate support is a roller while the right end of the beams can move vertically but the rotation is prevented there. The beams are referred to as FrsRp and PrsRp beams. Determination of the (eigenfrequencies) [critical loads] leads to three point eigenvalue problems associated with homogeneous boundary con- ditions. With the Green functions that belong to these eigenvalue problems we can transform them into eigenvalue problems governed by homogeneous Fredholm integral equations. The eigenvalue problems can then be reduced to algebraic eigenvalue problems that are solvable numerically by utilizing effective solution algorithms.