A New Static Analysis Approach for Free Vibration of Beams

This study deals with a new method for the free vibration analysis of beams under different boundary conditions. We show that it is possible to apply a static approach for solving free vibration systems, i.e., we obtain natural frequencies for free vibration of beams by analyzing static beam bending problems. Specifically, the basic governing equation for beams with harmonic loadings and resting on an elastic foundation is solved and the solutions are used directly to yield the beam free vibration solutions. In the free vibration analysis, the natural frequency can be a real number or an imaginary number while in the static analysis, the foundation stiffness can be either positive or negative. We show that one can solve the deflection of a beam subjected to a given concentrated force and subsequently deduce the possible infinite deflection when the stiffness becomes zero or negative. In such cases, there exists an equivalent relationship between the free vibration frequencies and the negative stiffness. Consequently, determining the natural frequencies becomes a problem of determining an appropriate negative foundation elastic constant. In general, the numerical vibration solutions can be obtained by analyzing the relationship between loadings and frequencies. For comparison, a comparison with the classical free vibration solutions is presented and excellent agreement is illustrated. We further show that this static approach for free vibration solutions has a clear edge over the classical free vibration approach in computational beam vibration solutions. Very accurate and convergent numerical solutions can be obtained using a very simple numerical solution method. This static approach for free vibration problems can be extended for plate, shell and other structural systems.