Natural frequencies of two cantilevers joined by a rigid connector at their free ends

The derivation of the equations of motion is given for a system consisting of two identical parallel cantilevers joined by a rigid connector at their free ends. Elementary beam theory is employed, and it is observed that the longitudinal and flexural deformations of the system are coupled through the boundary conditions but not through the differential equations. The associated free vibration problem is solved, and it is shown that the frequency determinant can be factored, yielding two independent frequency equations. One of these corresponds to the pure, free longitudinal motion of a pair of rods connected by a rigid body at their tips (both rods being equally stretched or compressed simultaneously), whereas the second and more complicated frequency equation is pertinent for the system undergoing flexural deformations in both rods, stretching in one rod, and compression in the other. This latter frequency equation is solved numerically, and the variations of the first six natural frequencies with connector thickness and length parameters are displayed graphically. Two orthogonality conditions for the eigenfunctions are derived, and a relatively simple form for the normalizing factor is presented.