Propagation of torsional waves in a circular elastic rod (original) (raw)
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Comparison of classical and modern theories of longitudinal wave propagation in elastic rods
16th International Congress on Sound and Vibration. Krakow, Poland, 5-9 July 2009 A unified approach to derivation of different families of differential equations describing the longitudinal vibration of elastic rods and based on the Hamilton variational principle is outlined. The simplest model of longitudinal vibration of the rods does not take into consideration its lateral motion and is described in terms of the wave equation. The more elaborated models were proposed by Rayleigh, Love, Bishop, Mindlin-Herrmann, and multimode models in which the lateral effect plays an important role. Dispersion curves, representing the eigenvalues versus wave numbers, of these models are compared with the exact dispersion curves of isotropic cylinder and conclusions on accuracy of the models are deduced. The Green functions are constructed for the classical, Rayleigh, Bishop, and Mindlin-Herrmann models in which the general solutions of the problem are obtained. The principles of construction of...
Mechanics of Composite Materials, 2008
Keywords: ini tial strain, in com press ible ma te rial, uni di rec tional fi brous com pos ite, wave dis per sion, wave prop a ga tion Within the frame work of a piecewise ho mo ge neous body model and with the use of the three-di men sional linearized the ory of elas tic waves in ini tially stressed bod ies (TLTEWISB), the prop a ga tion of axisymmetric lon gi tu di nal waves in a fi nitely prestrained cir cu lar cyl in der (fi ber) im bed ded in a fi nitely prestrained in fi nite elas tic body (ma trix) is in ves ti gated. It is as sumed that the fi ber and ma trix ma te ri als have the same den sity and are in com press ible. The stress-strain re la tions for them are given through the Treloar po ten tial. Nu mer ical re sults re gard ing the in flu ence of ini tial strains in the fi ber and ma trix on wave dis per sion are pre sented and dis cussed. These re sults are ob tained for the fol low ing cases: the fi ber and ma trix are both with out ini tial strains; only the fi ber is prestretched; only the ma trix is prestretched; the fi ber and ma trix are both prestretched si mul ta neously; the fi ber and ma trix are both precompressed si mul ta neously.
Compressional and torsional wave amplitudes in rods with periodic structures
The Journal of the Acoustical Society of America, 2002
To measure and detect elastic waves in metallic rods a low-frequency electromagnetic-acoustic transducer has been developed. Frequencies range from a few hertz up to hundreds of kilohertz. With appropriate configuration of the transducer, compressional or torsional waves can be selectively excited or detected. Although the transducer can be used in many different situations, it has been tested and applied to a locally periodic rod, which consists of a finite number of unit cells. The measured wave amplitudes are compared with theoretical ones, obtained with the one-dimensional transfer matrix method, and excellent agreement is obtained.
On extensional oscillations and waves in elastic rods
1998
Abstract The authors study the dispersive nature of propagating extensional waves in an infinitely long elastic rod within the framework of the linear theory of a Cosserat rod with two directors. The authors also identify certain material constants in the theory in a manner that is different from those used by others and consequently show that the resulting theory better captures the high-frequency dynamical behavior of three-dimensional rod-like bodies.
Some results on finite amplitude elastic waves propagating in rotating media
Acta Mechanica, 2004
Two questions related to elastic motions are raised and addressed. First: in which theoretical framework can the equations of motion be written for an elastic half-space put into uniform rotation? It is seen that nonlinear finite elasticity provides such a framework for incompressible solids. Second: how can finite amplitude exact solutions be generated? It is seen that for some finite amplitude transverse waves in rotating incompressible elastic solids with general shear response the solutions are obtained by reduction of the equations of motion to a system of ordinary differential equations equivalent to the system governing the central motion problem of classical mechanics. In the special case of circularly-polarized harmonic progressive waves, the dispersion equation is solved in closed form for a variety of shear responses, including nonlinear models for rubberlike and soft biological tissues. A fruitful analogy with the motion of a nonlinear string is pointed out.
After a brief historical survey of some work done on the linear theory of longitudinal vibrations and wave propagation in rods and tubes of uniform cross-section, a simple mathematical model for rods and tubes of linear elastic materials is proposed. Three suitably selected propagation modes (one extensional and two shear modes) with dispersion relations corresponding to mixed boundary conditions are coupled in order to approximately comply with zero-stress boundary conditions. The coupling gives a set of partial differential equations in the mode amplitudes, with time and a single space coordinate (along the axis of symmetry of the rod or tube) as independent variables. Then, the model is generalized to a set of partial integral-differential equations in order to be able to describe vibrations and wave propagation in rods and tubes made of linear hereditary-elastic solids. In this first part of the work, the focus is in either very low frequency or very high frequency phenomena using a simple model with only two coupled modes. The model allows a fairly elegant and comparatively powerful analytical approach to longitudinal vibrations and to longitudinal pulse propagation in solid waveguides. Analytical formulae for group velocities are derived, as well as asymptotic expressions for the propagation of mode amplitudes. The limitations and pitfalls of the model are assessed, and new experiments and digital simulations are suggested to test some of its predictions. Keywords: mechanical vibrations; wave propagation; elastic and hereditary-elastic materials; propagation modes in rods and tubes.
Addendum to" On Extensional Oscillations and Waves in Elastic Rods"
1998
In a previous paper on infinitesimal waves in an infinite elastic Cosserat rod [1], we proposed values for various material parameters, hereafter referred to as KB2. These parameters were determined by matching the response of the Cosserat rod with that of a three-dimensional elastic cylinder. Subsequently, we have studied extensional vibrations in rods of finite length [2]. For a small range of intermediate frequencies, it was found that a slightly different set of parameters yielded more accurate results.