Longitudinal wave propagation in a rod with variable cross-section

Abstract Vibration data are required for condition monitoring in machinery, and can only be collected indirectly after transferring through rods, shells, rotating shafts or other components in many engineering applications. Investigation on the transfer characteristics of vibration in these components is very helpful to guarantee the efficiency of the data collected indirectly. Here, the longitudinal wave propagation in a rod with variable cross-section is investigated. First, the equations of motion are established for the rod based upon the elementary wave theory, the Love theory and the Mindlin–Herrmann theory. Second, the transfer matrix method is employed to explore the propagation characteristics of the rod from the derived equations of motion. Finally, two kinds of rods with the cross-sections varying in the exponential and the polynomial forms are used to illustrate the analytical predictions of the propagation characteristics of the longitudinal wave, which are compared with the results from the finite element analysis (FEA) method. It is shown that Poisson's effect or the shear deformation plays a very important role in the longitudinal wave propagation in the rod and can widen the rod's stop band moderately. Moreover, the cut-off frequency of the rod is unconcerned with the variation form of the cross-section, but dependent on the area ratio between both the ends of the rod, even though Poisson's effect or shear deformation is included.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  Lothar Gaul,et al.  Numerical and experimental investigation of wave propagation in rod-systems with cracks , 2010 .

[3]  A. Paolone,et al.  Wave propagation in three-coupled periodic structures , 2007 .

[4]  Michael Y. Shatalov,et al.  COMPARISON OF CLASSICAL AND MODERN THEORIES OF LONGITUDINAL WAVE PROPAGATION IN ELASTIC RODS , 2009 .

[5]  Marek Krawczuk,et al.  Certain numerical issues of wave propagation modelling in rods by the Spectral Finite Element Method , 2011 .

[6]  A. Żak,et al.  A novel formulation of a spectral plate element for wave propagation in isotropic structures , 2009 .

[7]  Dikai Liu,et al.  Exact solutions for longitudinal vibration of rods coupled by translational springs , 2000 .

[8]  Shaopu Yang,et al.  Longitudinal waves in one dimensional non-uniform waveguides , 2011 .

[9]  W. Seemann.  Transmission and Reflection Coefficients for Longitudinal Waves Obtained by a Combination of Refined Rod Theory and FEM , 1996 .

[10]  Junhong Park,et al.  Transfer function methods to measure dynamic mechanical properties of complex structures , 2005 .

[11]  M. H. Li,et al.  Propagation constants of railway tracks as a periodic structure , 2007 .

[12]  Qiusheng Li,et al.  EXACT SOLUTIONS FOR FREE LONGITUDINAL VIBRATIONS OF NON-UNIFORM RODS , 2000 .

[13]  P. D. Folkow,et al.  Direct and inverse problems on nonlinear rods , 1999 .

[14]  Marek Krawczuk,et al.  Longitudinal wave propagation. Part II—Analysis of crack influence , 2006 .

[15]  Raman Sujith,et al.  EXACT SOLUTIONS FOR THE LONGITUDINAL VIBRATION OF NON-UNIFORM RODS , 1997 .

[16]  M. Toso,et al.  WAVE PROPAGATION IN RODS, SHELLS, AND ROTATING SHAFTS WITH NON-UNIFORM GEOMETRY , 2004 .

[17]  L. Cremer,et al.  Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies , 1973 .

[18]  Marek Krawczuk,et al.  Longitudinal wave propagation. Part I-Comparison of rod theories , 2006 .

[19]  Brian R. Mace,et al.  Wave propagation, reflection and transmission in non-uniform one-dimensional waveguides , 2007 .

[20]  Qiusheng Li.  Exact solutions for free longitudinal vibration of stepped non-uniform rods , 2000 .

[21]  Paresh Girdhar.  Practical Machinery Vibration Analysis and Predictive Maintenance , 2004 .

[22]  R. I. Sujith,et al.  Closed-form solutions for the free longitudinal vibration of inhomogeneous rods , 2005 .

[23]  Shuqi Guo,et al.  Wave motions in non-uniform one-dimensional waveguides , 2012 .

[24]  James F. Doyle,et al.  Wave Propagation in Structures , 1989 .