Special Aspects of Using Partial Response Method in Studying Vibration of Quasi-One-Dimensional Finite Element Models

BACKGROUND: Today, the finite element method (and software packages based on it, mainly foreign) is substantially the only computational method that actually allows solving parametrization problems of static deformation and vibration of complex structures. They often require much effort to prepare the inputs, advanced computing equipment, and funds to purchase the corresponding software packages. The development of an approach allowing to occasionally reduce the requirements and to relatively easily develop domestic computational software based on it seems quite relevant. AIM: To develop approaches that allow for less labor-intensive and faster parametrization of static deformation and associated flexural, longitudinal, and torsional steady-state oscillations (vibrations) of structures, allowing modeling by systems of quasi-one-dimensional finite element models. METHODS: The basic investigation method used in the study is a modification of the partial response method with some significant advantages in solving problems of steady-state forced vibration or static deformation of structures that allow modeling using quasi-one-dimensional finite element models. The proposed method — the discrete modification of the partial response method — consists in the recording of partial responses and recurrent partial parameters of algebraic equations to calculate the matrix entries. In this case, the conventional solution of the boundary value problem is replaced by the solution of a series of conjugation problems for pairs of partial systems, where each has its own boundary data. The number of pairs under consideration corresponds to the number of sections, for which the researcher wants to parametrize the studied process (amplitudes of linear and angular displacements, internal forces, and end reactions). The scope of problems to be solved is significantly expanded by using the method proposed by the authors to correct and/or modify the inertia and rigidity of elements of a quasi-one-dimensional finite element model and the external load acting on it. This approach allows us to build quasi-one-dimensional models for solving both one-dimensional and multidimensional problems and problems requiring to consider the additional factors that complicate the vibration pattern. RESULTS: The paper presents several applications of the discrete modification of the partial response method in combination with the correction and modification of the parameters of quasi-one-dimensional finite element models. CONCLUSION: The study allows us to conclude that the aim has been achieved, which is confirmed by the development of a discrete modification of the partial response method, allowing to calculate deformation parameters and steady-state vibrations for a certain class of structures.