Exact Tuning of a Vibration Neutralizer for the Reduction of Flexural Waves in Beams
In this manuscript, design of a vibration neutralizer for the reduction of flexural waves in beams is proposed. The structure considered is modeled as an euler-bernoulli beam experiencing a harmonically travelling wave. The neutralizer which consists of a mass, spring and damper (viscous or hysteretic) is attached at a point on the beam. Two designs are considered, in the first, the aim is the minimization of the transmitted energy past the neutralizer attachment point, and in the second, the aim is the maximization of the energy dissipated in the neutralizer. It is assumed that the excitation frequency content consists of a major frequency and some fluctuations around it. The neutralizer is tuned to the major excitation frequency, which is referred to as the tuning frequency. For minimum energy transmission, the transmissibility ratio is first defined in terms of dimensionless parameters. Then, it is shown that, the undamped neutralizer can be tuned to reflect all the energy back to its source at the tuning frequency. The tuning stiffness ratio is obtained analytically in terms of the tuning frequency and mass ratios. Furthermore, the neutralizer bandwidth is obtained in an analytical closed form, and it is shown that it increases with the increasing of the neutralizer mass. When damping is present in the neutralizer, the transmissibility ratio will no longer pass through a zero, instead it will exhibit a minimum value. Tuning of the neutralizer is achieved by forcing the minimum of the transmissibility ratio to coincide with the major excitation frequency. The tuning stiffness and damping ratios are obtained in closed form for both the viscous and hysteretic damping cases. For maximum energy dissipation, the tuned neutralizer can dissipate up to 50% of the incident energy. The maximum energy dissipated, i.e. 50% of the incident energy, is constant and occurs at the tuning frequency. It is neither affected by the neutralizer mass nor by the tuning frequency. The optimal design parameters, i.e. tuning stiffness and damping ratios are obtained analytically. It is shown that, the increasing of the neutralizer mass enhances the neutralizer performance in the vicinity of the tuning frequency. Finally, all analytical findings are verified numerically by a direct numerical resolution of the beam neutralizer equation of motion.