A closed-form study on the free vibration of a grid joined by a mass-spring system

This paper deals with the coupling flexural-torsional vibration analysis of a grid formed by two members. A mass-spring system is attached to the grid at the intersecting joint. The members of the grid are assumed to resist torsion as well as bending and shear. Moreover, the mechanical and geometrical properties of each member are different. In order to analyze the problem, a closed-form solution is obtained. In doing so, the governing differential equations of the system along with the pertinent boundary and compatibility conditions of the system are introduced. Then, the frequency parameters of the mechanical system under study are derived and given for the first five modes of vibration. Furthermore, some diagrams of the mode shapes of the grid are plotted. A finite element program is also written to verify the results. The results obtained from these two methods are in good agreement and the accuracy of our results is confirmed.