VIBRATION FREQUENCIES OF A ROTATING FLEXIBLE ARM CARRYING A MOVING MASS

A clamped–free flexible beam rotating in a horizontal plane and carrying a moving mass is modelled by the Euler–Bernoulli beam theory. The equation of motion is derived by Hamiltonʹs principle including the effects of centrifugal stiffening arising from the rotation of the beam. The motion of the moving mass and the beam is coupled. The equation of motion is a coupled non-linear partial differential equation where the coupling terms have to be evaluated at the position of the moving mass. In order to obtain the mode shapes which account for the motion of the moving mass, the solution is discretized into space and time functions and the beam is divided into two separate regions with respect to the moving mass. This results in two non-homogeneous linear mode shape ordinary differential equations with four boundary, one discontinuity and three continuity conditions. The power series method is used to solve for the mode shape differential equations. A frequency equation is derived giving the relationship between the non-dimensional modal frequencies and the four non-dimensional parameters, i.e., the moving mass position, the moving mass, the beam angular velocity and the total moment of inertia about the hub. The numerical bisection method is used to solve for the vibration frequencies under different parameters. Results are presented for the first three modes of vibration.