Natural characteristics for transverse vibration of Euler Bernoulli beams with variable end constraints

Abstract

Abstract The transverse vibration of Euler Bernoulli beam with mass of ends and springs is studied. The exact frequency equation is derived and natural frequencies and the corresponding mode shapes are calculated. With the linearly increasing mass of ends, natural frequencies and the rate of frequency change of the beam system initially decrease sharply and then level out, which demonstrates that the beam system is transforming from the free beam to the pinned beam. When the springs are added at two tips, the natural characteristics of the beam are affected by mass of ends and spring stiffness. If the added mass has much lower magnitude than that of the beam, the stiffness of springs exerts major impact on the increase of natural frequencies. While the added mass of ends is increased to the same magnitude of the beam, the natural characteristics of the beam are determined by both the mass of ends and spring stiffness. As the growing magnitude of added mass, mass of ends performs a dominant role in decreasing the natural frequencies. Therefore, spring stiffness and mass of ends should be first considered to establish different dynamic models accurately.