Equations Of Motion And Propagation Characteristics Of Elastic Waves Propagating In Rods And Rotating Shafts With Variable Cross-sections

This dissertation is partially supported by the National Natural Fund Project ’A new fatigue crack detection method for mechanical component based on laser ultrasonic’ (No.51375434) and the High Technology Research and Development Program of China’Research on the condition monitoring and fault diagnosis technique and its application software system of supercritical and ultra-supercritical steam turbine generator unit’ (No.2008AA04Z410), and focuses on the propagation characteristics of elastic waves propagating in the mechanical components such as rods and rotating shafts. The numerical simulation was adopted to study the propagation characteristics, and the result of which were validated by FEA as well as experiments result. The research helps to know the mechanism of elastic waves propagating in mechanical components such as rods and rotating shafts, and it’s of great significance to know how the vibration propagates.This study is organized as follows.The importance and significance of analyzing the propagation characteristics of the elastic wave propagating in rods and rotating shafts was discussed in Chapter 1. The development on studying the elastic wave propagation in rods and rotating shafts is summarized by the catogary of study object and content, solving method and applications. Research content which need be further studied and suitable method is selected herein. The content of this study is proposed at last.The classical elastic wave theories and other theories related were introduced and analyzed in Chapter 2. First, the concept and catogary of elastic waves was introduced as well as the corresponding wave types in rods and rotating shafts with uniform cross-section. The generalized Hamilton’s Principle was referred and by which several rod related theories as well as Euler-Bernoulii beam theory and Timoshenko beam theory were analyzed. An introduction about the propagation process was given out in the last, and the methods to derive and solve the transfer matrix were reviewed. And all of the above analysis is the basis of studying the elastic waves propagating in rods and rotating shafts with variable cross-section.Considering the influence of Poisson’s ratio and shear deformation, the equation of motion for a non-uniform rod was derived by adopting the elementary wave theory, the Love wave theory and the Mindlin-Hermann wave theory separately for the first time in Chapter 3. The transfer matrices were constructed accordingly by utilizing the separation principle to the equations of motion. The effect of Poisson’s ratio and shear deformation, as well as variation forms and the area ratio of the two ends, were valuated in detail, and the conclusion is that: Poisson’s ratio and shear deformation can’t be neglected, and there is no difference found between the two variation forms, while a higher area ratio of the two ends will cause a higher cut-off frequency.A’Modeling by Parts’method was utilized to deduce the transfer matrix for a multi-step rod with variable cross-section in Chapter 4. The transfer matrix for the joint is derived by considering the match conditions of a joint between two adjacent single rods, as well as the possible elastic constraint. The transfer matrix of the whole multi-step rod with non-uniform cross-section could be deduced by combing the transfer matrix of a single rod. A very close study to the propagation characteristics of the longitudinal wave propagating in a multi-step rod reveals that the influence of Poisson’s ratio and shear deformation can’t be neglected, the external elastic force at the joint will change the stop bands and a higher discontinuity of the cross-section will raise the first stop band to a higher level.The equation of motion for a single rotating shaft with variable cross-section was constructed based on the Timoshenko beam theory and the relationship between the shaft coordinates and the Cartesian coordinates by applying the generalized Hamilton’s Principle in Chapter 5. And then the transfer matrix for the shaft was derived from the motion equation by utilizing the separation principle. After analyzing the variation forms and the area ratio, a conclusion could be made that the variation form of the shaft has no influence on the propagation characteristics while a higher area ratio of the two ends will reduce the stop band to a lower level but will widen the stop band, meanwhile, increasing the rotating speed will lead to the result as increasing the area ratio did.A’ Modeling by Parts’ method was utilized to study a multi-step rotating shaft with a variable cross-section and emerging a transverse crack in Chapter 6. The transfer matrix of the crack was studied first by studying the boundary conditions of it. After that, the transfer matrix for the whole cracked rotating shaft was deduced by combing the transfer matrix of a single rotating shaft and that of a crack. A detailed study found that the depth of the crack and the rotating speed may affect the propagation characteristics. A new stop band will appear in the lower range and the width of which will be widen as the increase of the depth of the transverse crack. And the first stop band will be reduced to a lower level as the increasing of the rotating speed of the rotating shaft.Conclusions of this study is summarized in Chapter 7, and in which some advanced topics for future work could be found also.