Combining parametric discreteness and continuity for analyzing dynamic models of structures

Abstract

This paper describes a method for building dynamic models that contain rod bending elements with distributed and concentrated inertial and stiffness parameters, and their analysis based on the harmonic element method. As a rule, the vibration effects of structures are calculated on the basis of mass discretization, although the application of such methods entails certain difficulties.

Discrete models are considered to be a priori approximations with limited possibilities of error estimation. The dynamic parameters of the model vary depending on its dimensionality as well as on the transformation methods. Numerical results with arrays and matrices of high dimensionality make it difficult to analyze and evaluate the calculation results. Therefore, structural calculations for stationary dynamic effects based on the use of elements with distributed and concentrated masses prevent the above-mentioned consequences of full discretization. However, such discrete-continuum (hybrid) dynamic models require the sewing of heterogeneous elements at the formation stage. In addition, some complications occur when solving these combined systems containing ordinary differential equations and partial differential equations. These issues can be solved by using the author's harmonic element method, implementing the nodal sewing of heterogeneous elements, as well as providing solutions as amplitudes of oscillations of the combined model nodes along certain necessary directions. The specified features of the proposed method allow us to identify it as a separate class with the name of the harmonic element method.