Method of reference problems for obtaining approximate analytical solution of multiparametric Sturm–Liouville problems

Approximate analytical formulas are obtained for the eigenfrequencies of longitudinal oscillations of an elastic rod with different mechanical fixings of the ends. The eigenfrequencies are found by solving Sturm–Liouville problems with the third kind boundary conditions as roots of transcendental equations. Homogeneous boundary conditions contain one or more parameters whose values are calculated through the indices of mechanical system. Approximate expression for analytical dependencies of the eigenfrequencies on the single parameter are obtained for one-parametric problems, which are called reference ones. We propose a method for obtaining approximate analytical expression for dependencies of the eigenfrequencies on several parameters in boundary conditions by sequentially solving the reference problems. The two-parametric Sturm–Liouville problem is solved by the proposed method.