Free Vibration Analysis of a Sloping-frame: Closed-form Solution versus FiniteElement Solution and Modification of the Characteristic Matrices

This article deals with the free vibration analysis and determination of dynamic characteristics of asloping-frame. First of all, a closed-form solution is proposed and then, a numerical analysis isperformed for some verification purposes. The closed-form solution is developed by solving the frameequations of motion, directly. For this reason, some mathematical techniques are utilized, such asFourier transform and the well-known complementary solutions. In this way, some differentialequations must be solved, and several boundary conditions should be satisfied. Herein, the moreaccurate derivation of one of twelve boundary conditions is the most important challenge of this paper.This boundary condition is expressed as three distinctive versions, and the free vibration parameters ofthe frame for the three versions are obtained. Moreover, these results are obtained by the use of thefinite element method. In this comparison process, some differences are observed between the closedformand the numerical results. This fact motivated us to propose some modifications in thecharacteristic matrices of the finite element model of the frame. This modification makes the results ofthe finite element method similar to the results of the first version of the closed-form solution. Finally,the natural frequencies and mode shapes are presented for a wide range of angles of the slopingmember.