The features of solutions of dynamical systems with an external exciting load and non-conservative dynamical systems with pairwise interaction of degrees of freedom

2021 ◽  
Author(s):  
B.D. Kashfutdinov ◽  
A.F. Georgiev
Keyword(s):  
Dynamical Systems ◽  
Analytical Solution ◽  
Degrees Of Freedom ◽  
Pair Interaction ◽  
Pairwise Interaction ◽  
Mechatronic Systems ◽  
Automated Control ◽  
Finite Element Software ◽  
Harmonic Action ◽  
Apparent Similarity

When mathematical models are developed, as a rule, a number of assumptions is done, which makes it possible to simplify the model, reduce its dimension and simulation time, or use the dimension reduction method. When modeling non-conservative systems with pairwise interaction of degrees of freedom, e.g. mechatronic systems, an elastic aircraft in a flow, an aeroelastic aircraft with an automated control system, etc., there is a desire to reduce the problem to a conservative dynamic system with harmonic action. The study shows that despite the apparent similarity of the tasks, they have significant differences that cannot be ignored. Differences in the behavior of conservative dynamical systems and non-conservative dynamical systems with pair interaction of degrees of freedom are considered. The results are demonstrated on the simplest example with an analytical solution, and in the finite element software package MSC.Nastran. The results of the solution in MSC.Nastran are compared with the results of the analytical solution.

HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2823-2869 ◽  
Author(s):  
Z. E. MUSIELAK ◽  
D. E. MUSIELAK
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
General Rules ◽  
Van Der Pol Oscillators ◽  
Control And Synchronization

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

2013 ◽  
Vol 23 (03) ◽  
pp. 1330009 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
MOZHDEH S. FARAJI MOSADMAN
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Eigenvalue Analysis ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

Effect of a Planetary Gearbox on the Dynamics of a Rotor System

2018 ◽  
Author(s):  
Ali Tatar ◽  
Christoph W. Schwingshackl
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Behaviour ◽  
Rotor System ◽  
Mode Shapes ◽  
Rotating System ◽  
Planetary Gearbox ◽  
Finite Element Software ◽  
Gyroscopic Effects ◽  
The Masses ◽  
Parameter Values

The dynamic analysis of rotors with bladed disks has been investigated in detail over many decades and is reasonably well understood today. In contrast, the dynamic behaviour of two rotors that are coupled via a planetary gearbox is much less well understood. The planetary gearbox adds inertia, mass, stiffness, damping and gyroscopic moments to the system and can strongly affect the modal properties and the dynamic behaviour of the global rotating system. The main objective of this paper is to create a six degrees of freedom numerical model of a rotor system with a planetary gearbox and to investigate its effect on the coupled rotor system. The analysis is based on the newly developed finite element software “GEAROT” which provides axial, torsional and lateral deflections of the two shafts at different speeds via Timoshenko beam elements and also takes gyroscopic effects into account. The disks are currently considered as rigid and the bearings are modelled with isotropic stiffness elements in the translational and rotational directions. A novel planetary gearbox model has been developed, which takes the translational and rotational stiffness and the damping of the gearbox, as well as the masses and inertias of the sun gear, ring gear, planet gears and carrier into account. A rotating system with a planetary gearbox has been investigated with GEAROT. The gearbox mass and stiffness parameters are identified as having a significant effect on the modal behaviour of the rotor system, affecting its natural frequencies and mode shapes. The higher frequency modes are found to be more sensitive to the parameter changes as well as the modes which have a higher deflection at the location of the gearbox on the rotor system. Compared with a single shaft system, the presence of a gearbox introduces new global modes to the rotor system and decouples the mode shapes of the two shafts. The introduction of a planetary gearbox may also lead to an increase or a reduction of the frequency response of the rotor system based on gear parameter values.

Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Buckling and Nonlinear Dynamic Examples

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Karin Nachbagauer ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Continuum Mechanics ◽  
Degrees Of Freedom ◽  
Mass Matrix ◽  
Finite Element Software ◽  
High Order Convergence ◽  
Dynamics Problems ◽  
Transverse Slope ◽  
Shear Deformable

For the modeling of large deformations in multibody dynamics problems, the absolute nodal coordinate formulation (ANCF) is advantageous since in general, the ANCF leads to a constant mass matrix. The proposed ANCF beam finite elements in this approach use the transverse slope vectors for the parameterization of the orientation of the cross section and do not employ an axial nodal slope vector. The geometric description, the degrees of freedom, and a continuum-mechanics-based and a structural-mechanics-based formulation for the elastic forces of the beam finite elements, as well as their usage in several static problems, have been presented in a previous work. A comparison to results provided in the literature to analytical solution and to the solution found by commercial finite element software shows accuracy and high order convergence in statics. The main subject of the present paper is to show the usability of the beam finite elements in dynamic and buckling applications.

Automatic Verification of Modeling Rules in Systems Engineering for Mechatronic Systems

2013 ◽  
Author(s):  
Lydia Kaiser ◽  
Roman Dumitrescu ◽  
Jörg Holtmann ◽  
Matthias Meyer
Keyword(s):  
Systems Engineering ◽  
Degrees Of Freedom ◽  
System Model ◽  
Object Constraint Language ◽  
Mechatronic Systems ◽  
Control Engineering ◽  
Holistic View ◽  
Constraint Language ◽  
Software Engineers ◽  
Effective Product

Mechatronics is the close interaction of mechanics, electronics, control engineering and software engineering. The increasing complexity of mechatronic systems results in a challenging development process and particularly requires a consistent comprehension of the tasks between all the engineers involved. Especially during the early design phases, the communication and cooperation between the mechanical, electrical, control and software engineers is necessary to establish a basis for efficient and effective product development. The approach of Model-Based Systems Engineering focuses on this aspect by means of an abstract but superordinate system model. It enables a holistic view of the system. The system model can be specified using the Systems Modeling Language (SysML). The language allows many degrees of freedom to specify a fact, bearing in mind that different system architects can specify the same fact in different ways. This leads to system models that can be interpreted in many ways. Thus, these models are hard to consistently compare and interpret, resulting in communication issues. In order to tackle this problem, we present a concept that uses modeling rules supporting model comparability. We formalize them by means of checks implemented in the programming language Java and the Object Constraint Language (OCL) in order to automatically verify the system model’s compliance with these rules.

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