Dynamic behavior and modal control of beams under moving mass
The constitutive equation of an Euler–Bernoulli beam under the excitation of moving mass is considered. The dynamics of the uncontrolled system is governed by a linear, self-adjoint partial differential equation. A Dirac-delta function is used to describe the position of the moving mass along the beam and its inertial effects. An approximate formulation to the problem is obtained by limiting the inertial effect of the moving mass merely to the vertical component of acceleration. Having defined a “critical velocity” in terms of the fundamental period and span of the beam, it is shown that for smaller velocities, the approximate and exact approaches to the problem almost coincide. Since, the defined critical velocity is fairly large compared to those in practical cases, the approximate approach can effectively be used for a wide range of problems. There is however a slight variation in critical velocity depending on the weight of the moving mass. On the other hand, it is shown that the effect of higher vibrational modes is not negligible for certain velocity ranges. Finally, a linear classical optimal control algorithm with a time varying gain matrix with displacement-velocity feedback is used to control the response of the beam. The efficiency of the control algorithm in suppressing the response of the system under the effect of moving mass with different number of controlled modes and actuators is investigated.
Fayaz Rahimzadeh Rofooei
Mohammad Reza Shadnam