Stability analysis of mechanical systems with distributed delay via decomposition

The article analyzes a linear mechanical system with a large parameter at the vector of velocity forces and a distributed delay in positional forces. With the aid of the decomposition method, conditions are obtained under which the problem of stability analysis of the original system of the second-order differential equations can be reduced to studying the stability of two auxiliary first-order subsystems. It should be noted that one of the auxiliary subsystems does not contain a delay, whereas for the second subsystem containing a distributed delay, the stability conditions are formulated in terms of the feasibility of systems of linear matrix inequalities. To substantiate this decomposition, the Lyapunov direct method is used. Special constructions of Lyapunov—Krasovskii functionals are proposed. The developed approach is applied to the problem of monoaxial stabilization of a rigid body. The results of a numerical simulation are presented confirming the conclusions obtained analytically.

Stability analysis of gyroscopic systems with discrete and distributed delays

This paper provides stability analysis results for a linear mechanical system with a large parameter at the vector of gyroscopic forces and with delay in positional forces. Both cases of discrete and distributed delay are studied. Using the decomposition method and Lyapunov–Krasovskii functionals, conditions are found under which delay does not disturb the asymptotic stability of the considered system. The effectiveness of the obtained results is illustrated by a simulation example.

A Novel 4D Chaotic System Based on Two Degrees of Freedom Nonlinear Mechanical System

In this study, a non-linear mechanical system with two degrees of freedom is considered in terms of chaos phenomena and chaotic behaviour. The mathematical model of the system was moved to the state space and presented as a four dimensional (4D) chaotic system. The system’s chaotic behaviour was investigated by performing dynamic analyses of the system such as equilibria, Lyapunov exponents, bifurcation analyses, etc. Also, the electronic circuit realisation is implemented as a real-time application. This system exhibited vibration along with noise-like behaviour because of its very low amplitude values. Thus, the system is scaled to increase the amplitude values. As a result, the electronic circuit implementation of the 4D chaotic system derived from the model of a physical system is realised.

Targeted energy transfer in a 2-DOF mechanical system coupled to a non-linear energy sink with varying stiffness

The dynamical behavior of a non-linear mechanical system with two degrees of freedom (DOFs) during free and forced excitations is studied analytically and numerically. The non-linearity of the system is represented intentionally by a smooth non-linear simple function with periodically varying stiffness around a constant value for the sake of practical investigations. Analysis of the system leads to a method that could be used to design the non-linear energy sink (NES) so that the behavior of the system during relaxation and its strongly modulated response (SMR) could be improved versus the constant stiffness configuration.

Simultaneous Output-Only Identification of Physical Parameters and Unknown Inputs for Linear Mechanical Systems

This paper has studied the identification problem of linear mechanical systems where inputs are unknown and only displacement data are accessible for measurement. Eigensystem Realization Algorithm (ERA) has been used along with physical constraints considerations in time domain to simultaneously identify two separate models for the physical system and the unknown inputs. Inputs are assumed to be an arbitrary combination of harmonic signals with frequencies higher than natural frequencies of the physical system by which a linear mechanical system is meant in this paper. Adding physical constraints and utilizing canonical real Jordan form of the identified system leads to a unique analytical solution. To validate the theory part, a set of simulations has been run that demonstrates the physical parameters and input model can be estimated accurately.