On the Analysis of Longitudinal Vibrations and the Stress State of Composite Rods with Elastic Shear Bonds

A differential equation of free longitudinal vibrations of a composite rod of circular cross-section is obtained, the outer part 1 of which is connected to the inner part 2 by means of elastic shear bonds. The general solution of the obtained equation of the fourth order is constructed. The scheme of transition to the solution of similar problem in the case of prismatic symmetric composite system is marked. For the independent analysis of vibrations of parts 1 and 2, separate solutions are constructed by means of proper transformation of the integration constants in the general solution. Dependencies for the determination of displacements (vibration modes) and frequency equations for a number of cases of fixation of a composite rod (free or rigidly fixed at the ends), free inner part 2 and outer part 1 rigidly fixed at the ends are given. The dependence of the sought quantities on the stiffness of shear bonds, determined by the parameter μ, is noted. The frequencies are calculated, and for the case of mixed fixation of parts 1 and 2, the fundamental modes of natural vibrations and corresponding graphic dependences for normal and tangential stresses are constructed. It is shown that to determine the values of the parameter μ it is necessary to experimentally determine the natural (resonant) vibration frequency. The scheme of methodology for the determination of μ is outlined. The graphic dependence of μ on the fundamental frequency for the case of mixed fixation is given. It is shown that with decreasing μ, the eigenfrequency increases. The practical importance of this phenomenon for testing composite systems is marked. Based on the analysis of displacements, normal and tangential stresses was established that the ratios of the extreme values of displacements and stresses are also approximately 10. It was concluded that the stress-strain state of the inner free end part 2 of the composite rod practically does not depend on the rigidity of shear bonds under fundamental frequency vibrations. It was concluded that the results obtained and their analysis can serve as a basis for the experimental determination of adhesion strength under variable loading. The desired characteristic here is the limit cyclic tangential stresses acting between the layers.