Corrections for poisson effect in longitudinal vibrations and shearing deformations in transverse vibrations applied to a prismatic orthotropic body
This chapter deals with elastic wave propagation in a finite orthotropic bar. In the frrst part, the effect of lateral motion caused by the Poisson contraction phenomenon is discussed. This phenomenon induces wave dispersion and consequently vibration is considered as "quasi-longitudinal", rather than pure longitudinal displacement of matter along the bar. Several authors have proposed analytic solutions to quasi-longitudinal wave propagation in rods. Solutions for isotropic bodies with a circular cross section have been discussed. The analytic solution presented here applies to an orthotropic rod with a rectangular cross-section. This solution is based on energy considerations. The material axes are collinear to those of the prismatic bar. The second part of this note is focused on the effects of rotary inertia and shearing deformations which occur during transverse vibrations of prismatic bars. The classical approach assumes that the cross-sectional dimensions of a bar are small in comparison with its length. To take into account these effects, corrections were made using a numerical factor (associated to the shearing force) to obtain a more complete differential equation of motion. A novel method is proposed to provide the exact solution of this differential equation for a rod of rectangular crosssection constituted by an orthotropic body in free-free conditions. This method uses a specific type of musical sound synthesis modeling coupled with classic considerations of elastic behavior of orthotropic materials.
| Summary: | This chapter deals with elastic wave propagation in a finite orthotropic bar. In the frrst part, the effect of lateral motion caused by the Poisson contraction phenomenon is discussed. This phenomenon induces wave dispersion and consequently vibration is considered as "quasi-longitudinal", rather than pure longitudinal displacement of matter along the bar. Several authors have proposed analytic solutions to quasi-longitudinal wave propagation in rods. Solutions for isotropic bodies with a circular cross section have been discussed. The analytic solution presented here applies to an orthotropic rod with a rectangular cross-section. This solution is based on energy considerations. The material axes are collinear to those of the prismatic bar. The second part of this note is focused on the effects of rotary inertia and shearing deformations which occur during transverse vibrations of prismatic bars. The classical approach assumes that the cross-sectional dimensions of a bar are small in comparison with its length. To take into account these effects, corrections were made using a numerical factor (associated to the shearing force) to obtain a more complete differential equation of motion. A novel method is proposed to provide the exact solution of this differential equation for a rod of rectangular crosssection constituted by an orthotropic body in free-free conditions. This method uses a specific type of musical sound synthesis modeling coupled with classic considerations of elastic behavior of orthotropic materials. |
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