On the Countering of Free Vibrations by Forcing: Part I—Non-Resonant and Resonant Forcing with Phase Shifts

The question addressed is whether the free oscillations of a continuous system can be suppressed, or at least the total energy reduced, by applying external forces, using as example the linear undamped transverse oscillations of a uniform elastic string. The non-resonant forcing at an applied frequency, distinct from all natural frequencies, does not interact with the normal modes, whose energy is unchanged, and adds the energy of the forced oscillation, thus increasing the total energy, that is the opposite of the result being sought. The resonant forcing at an applied frequency, equal to one of the natural frequencies, leads to an amplitude growing linearly with time, and hence the energy is growing quadratically with time, implying an increase in total energy after a sufficiently long time. A reduction in total energy is possible over a short time, say over the first period of oscillation, by optimizing the forcing. In the case of a concentrated force, by optimizing its magnitude and location, the total energy with forcing in one period is reduced by a modest maximum of 2% relative to the free oscillation alone. The conclusion is similar for several concentrated forces. In the case of a continuously distributed force, by optimizing the spatial distribution, it is possible to reduce the energy of the total oscillation to one-fourth of that of the free oscillation over the first period of vibration. This shows that continuously distributed forces are more effective at vibration suppression than point forces.