NATURAL FREQUENCIES AND OPE–LOOP RESPONSES OF AN ELASTIC BEAM FIXED ON A MOVING CART AND CARRYING AN INTERMEDIATE LUMPED MASS

In this paper, the motion of a Bernoulli–Euler cantilever beam clamped on a moving cart and carrying an intermediate lumped mass is considered. The equations of motion of the beam–mass–cart system are analyzed through unconstrained modal analysis, and a unified characteristic equation for calculating the natural frequencies of the system is established. The changes of natural frequencies and the corresponding mode shapes with respect to the changes in the ratios between the beam mass, the lumped mass and the cart mass and to the concentrated position of the lumped mass are investigated with the frequency equation, which can be generally applied to this kind of system. The exact and assumed-mode solutions including the dynamics of the base cart are obtained, and the open-loop responses of the system by arbitrarily designed forcing functions are given by numerical simulations. The results match well with physical phenomena even in the extreme cases where the mass is attached to the bottom and to the top of the beam.