Steady‐State Responses of One‐Dimensional Periodic Flexural Systems

Expressions are derived that describe the steady‐state behavior of infinitely long beams with uniformly spaced attached impedances and that permit damping and fluid loading to be taken into account readily. The case where the attached impedances produce only lateral forces is treated on the basis of classical beam‐flexure theory. Relations governing the segment‐to‐segment propagation constant and the impedance the system presents to a force acting at an impedance attachment point are derived; the response to forces acting at all impedance attachment points, with a constant phase difference between adjacent forces, is also deduced. Additional results, applicable only in the region outside the near fields of the impedance attachment points, in cases where the interval between attached impedances encompasses many flexural wavelengths, are formulated in terms of the reflection and transmission coefficients that are effective at the impedance attachment points, and thus apply for impedances of any degree of complexity. A corresponding expression for the propagation constant is obtained, and a method (which is suitable also for numerical analysis of a wide variety of complex one‐dimensional systems) is developed for determining the steady‐state oscillations resulting from a given wave or waves injected at a given location.