On the Dynamical Problems of Linear Dissipative Mechanical Systems Under Natural Vibrations

Abstract

This study investigates the dynamic behavior of linear dissipative mechanical systems subjected to natural vibrations, with particular emphasis on the influence of viscoelastic dissipation mechanisms. The research develops mathematical models for analyzing the free vibration response of mechanical systems composed of rigid and deformable bodies, taking into account the hereditary properties of viscoelastic materials. The dissipative effects are represented through additional integral terms appearing in the governing integro-differential equations, where hereditary kernels characterize the time-dependent material behavior and energy dissipation processes. Based on the principles of analytical mechanics and the application of Lagrange's equations, a comprehensive formulation of the equations of motion is established. The resulting mathematical model enables the determination of characteristic exponents, damping coefficients, and eigenfrequencies of the system. To improve the efficiency and accuracy of the computational procedure, modified versions of the Gauss and Muller numerical methods are proposed. Furthermore, an algorithm employing complex arithmetic is developed for solving the characteristic equations associated with viscoelastic systems and for identifying the corresponding complex eigenvalues. The stability and asymptotic behavior of the oscillatory processes are analyzed through the obtained characteristic roots. Numerical simulations are carried out for representative mechanical systems, and the influence of viscoelastic parameters and hereditary kernels on vibration attenuation, frequency variation, and dynamic stability is investigated. The results demonstrate the effectiveness of the proposed computational approach in accurately predicting the dynamic response of dissipative systems and provide valuable insights into the design and analysis of engineering structures incorporating viscoelastic materials. The developed methodology can be applied to a wide range of engineering problems involving vibration control, structural dynamics, and the assessment of long-term behavior in mechanical systems with energy dissipation mechanisms.