Abstract

The dynamic behavior of discrete periodic one-dimensional structures is approached by considering transverse vibrations of structures made of repeated unbraced frames. Assuming the frame size is small compared to the modal wavelength, equivalent macroscopic beam descriptions are obtained by the homogenization method of periodic discrete media. The macroscopic parameters are expressed as functions of the mechanical and geometrical properties of the frame elements.

Depending on the order of magnitude (relative to the scale ratio) of the shear force, the global bending and the inner bending, four families of beams are shown to be possible. A generic beam governed by a differential equation of the sixth degree is shown to encompass all the other types.

Simple criteria are established to identify the relevant model for real structures. A comparison of these theoretical results with numerical modeling is satisfactory even in the case of weak scale separation. In fact, an investigation of the higher orders terms shows that zero order descriptions are valid up to the second order. Lastly, analogies with micromorphic media are discussed.

Keywords

discrete structure, modal analysis, beam theory, homogenization, micromorphic media

Authors