Damped vibration of a cracked Timoshenko beam with ends supported with damper, linear and rotational springs is investigated. Frequencies in complex forms have been obtained for both cracked Euler-Bernoulli and Timoshenko beams. Depending upon the crack-depth and crack-location, frequencies have been tabulated in each case. The results have also been compared in terms of the ratio of the beam depth to the beam length. Modal shapes for various conditions have also been plotted.

In this study, we propose four methods for the analytical solution of second order ordinary non-homogeneous differential equations with variable coefficients. In the first case, the method directly gives the solution in explicit or integral form. In the second method, the solution of the problem reduces to the solutions of adjoined second order ordinary differential equation of homogeneous type. As long as the analytical solution of two adjoined equations can be solved, the analytical solution can always be found. In the third method, the differential equation is transformed into Riccati equation. Riccati equation is solved by means of a method recently developed. In order to solve non-homogeneous differential equation, the fourth method is developed. The strategy is different but the solution is again based on the solution of Riccati equation.

In this study, the longitudinal vibration of rods with variable cross-sections is studied. For the analytical solution of the problem, a new analytical method based on a recently developed method on the Riccati differential equation is utilized. The governing equation is reduced to Hill’s type second-order ordinary differential equation. The transformed equations can readily be solved analytically for various cases according to the method. Seven cases have been considered, and the frequency equations for each case have been obtained. According to the method developed, the problem is solved in the simplest way. By using the present method, the reader can readily decide whether the problem is solved analytically or numerically. The present method can solve the problem of longitudinal vibration of rods having cross-sections of arbitrary shape. Finally, the method is also applied to the longitudinal vibration of stepped rods. Mode shapes are plotted for special values.