Free Vibration Analysis of a Sloping-frame: Closed-form Solution versus Finite Element Solution and Modification of the Characteristic Matrices (TECHNICAL NOTE)

This article deals with the free vibration analysis and determination of dynamic characteristics of a sloping-frame. First of all, a closed-form solution is proposed and then, a numerical analysis is performed for some verification purposes. The closed-form solution is developed by solving the frame equations of motion, directly. For this reason, some mathematical techniques are utilized, such as Fourier transform and the well-known complementary solutions. In this way, some differential equations must be solved, and several boundary conditions should be satisfied. Herein, the more accurate 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 of the frame for the three versions are obtained. Moreover, these results are obtained by the use of the finite element method. In this comparison process, some differences are observed between the closedform and the numerical results. This fact motivated us to propose some modifications in the characteristic matrices of the finite element model of the frame. This modification makes the results of the 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 sloping member.