Locally periodic Timoshenko rod: Experiment and theory

Abstract

The flexural vibrations of a locally periodic rod, which consists of N unit cells, are discussed both from the experimental and theoretical points of view. Timoshenko's beam theory and the transfer matrix method are used to calculate the normal-mode frequencies and amplitudes. The theoretical values are then compared with the experimental ones, which are obtained using an electromagnetic acoustic transducer (EMAT). Good agreement between the numerical results and the experimental measurements is obtained. It is shown that as N grows, a band spectrum emerges.

Experimental study of the Timoshenko beam theory predictions

“…Almost 90 years ago, Timoshenko Beam Theory (TBT) was established [3]. This theory agrees with the Bernoulli-Euler results for the lower normal modes but it fits experimental data at higher frequencies as it is well known and we have proved experimentally for a rod with free-free boundary conditions [4].…”

“…For values of the frequency f o f c , TBT seems to agree well with theoretical and numerical calculations [7][8][9][10] as well as with experimental measurements of the normal-mode spectrum [4,11,12]. However, for f 4f c this is no longer the case, and what was called the second spectrum of TBT has been the subject of a long standing debate [11,[13][14][15][16][17][18][19][20][21][22][23].…”

“…(1) and (2) with the boundary conditions (13) and (14). One should note that the boundary conditions (20) and (23) depend on the frequency and that they agree with the Bernoulli-Euler boundary conditions [1,4,12] for low frequencies.…”

“…In this subsection we study a rod of length L with free-free boundary conditions since these boundary conditions can be easily obtained in the laboratory [4,12]. The boundary conditions for a beam with free ends are given by the vanishing of moments and shear forces [5,26], qC qz…”

Building and destroying symmetry in 1-D elastic systems

“…They have recently been found in optics as well [3,4]. As will be shown, by modifying appropriately the small rods in a systematic way one can obtain an elastic analog of WSL [11].…”