Exact tuning of a vibration neutralizer for the reduction of flexural waves in beams

In this work, designs of vibration neutralizers for the reduction of flexural waves in beams are proposed. The system considered consists of 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, and in the second, the aim is the maximization of the energy dissipated in the neutralizer. For minimum energy transmission, the undamped neutralizer can be tuned to reflect all the energy back to its source at the tuning frequency. The tuning stiffness ratio and the neutralizer bandwidth is obtained in an analytical closed form. For maximum energy dissipation, the 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 tuning parameters are obtained analytically. Finally, in all the above cases, the increasing of the neutralizer mass enhances the neutralizer performance. All analytical findings are validated numerically.