On the influence of a concentrated inclusion on the spectrum of natural vibrations of a string and Bernoulli-Euler beam

Abstract

The results of a study of small transverse vibrations of a string and Bernoulli-Euler beam with a concentrated inclusion are presented. The physical properties of the string and the beam are assumed to be constant, the inclusion is modeled using the Dirac delta function and described by two parameters: location and mass. The problem of determining these parameters by measuring the shift of the resonant frequency is considered. The basic method is the eigenfunction expansion of displacement. Expansion coefficients are determined using the Greenberg method. Their substitution into the original expansion in the case of a point defect allows us to obtain a characteristic equation that determines the effect of inclusion on the string and beam natural frequencies. An analytical solution to the problem of small transverse vibrations of a string and Bernoulli-Euler beam with a point inclusion is presented. A method for possesing frequency equations that completely determine the influence of inclusion on the oscillation spectrum is proposed. Basing on the proposed method, expressions for identifying the inclusion parameters are derived, and the dependences of these parameters on the resonant frequency shift are presented. The possibility of independently determining the mass and location of the defect by measuring the shift of two natural frequencies is shown. The work is aimed at developing analytical methods for modeling the dynamics of continuum mechanical systems with a heterogeneous structure. The description of their dynamic response is of significant practical interest for creating various types of sensors, such as accelerometers, speed sensors, pressure sensors and others. The results obtained in this article can be used in the elaboration of mass detectors, the operation of which is based on changes in the natural frequency of oscillations.