FORCED OSCILLATIONS AND RESONANCE OF INFINITE PERIODIC STRINGS

Abstract An infinite string, supported by the equidistantly spaced identical suspensions, is considered. Each suspension consists of a spring and a dashpot with viscous damping, in parallel. The small transverse oscillations of the string are affected by the viscous drag of an external medium. The concentrated harmonic force, moving steadily along the string, causes steady-state oscillations. This means that the displacement along the string across the suspension spacing brings the time delay and the phase lead in the string's transverse deflection. The former is equal to the time of the exciting force motion over the spacing. The latter is equal to the change of the exciting force phase over this time. The steady-state nature of the oscillations and the linear property of the infinite periodic structure allows one to consider just one segment of the string between the neighbouring suspensions. The string deflection is governed within the segment by a partial differential equation and two boundary conditions that include the time delay and the phase lead. Both approach infinity as the speed of exciting force approaches zero and so stationary excitation cannot be directly included in the present consideration. It is supposed that the string segment had been at rest before the force approached and returned to rest after the force had moved away. Fourier transformation is used to solve the boundary problem in the infinite strip. The solution is represented in the form of a single integral that is as good for calculation as for qualitative analysis. These show that the periodic string resonance takes place, if any viscous resistance is absent and the integrand has a real pole of the second order. The string oscillations' dependence on the suspension stiffness is studied. If the stiffness is small enough, then a Doppler effect takes place. Two limit cases which correspond to the stiffness that approaches zero or infinity are considered. In the first case, the string is free. In the second one, the rigid suspensions divide the string into an infinite sequence of isolated segments. Any isolated segment is exposed to the moving exciting force over a limited time. Therefore, string resonance is impossible. The integrand has no second order real pole as well. In order to include stationary excitation into consideration, a suitable limit procedure is used. Resonance in response to such an excitation is studied and resonant frequencies are found. If the excitation point coincides with the suspension location or with the string mid-span, then the exciting force produces symmetric oscillations in the string. In these particular cases of excitation, resonance at some resonant frequencies disappears, but anti-resonance, that seems to be impossible in response to moving excitation, appears instead. In some cases, the exciting force produces a standing wave in the string, each suspension coincides with the wave node and so any suspension is strictly fixed as well as the excitation point. Such anti-resonance is not affected by suspension viscous damping. In other cases, if viscous damping is small, the excitation point and suspensions experience small oscillations.