On Degeneracy of Eigenvalues and Recursive Solution of Symmetrically Coupled Dynamic Systems

1970 ◽  
Vol 37 (4) ◽  
pp. 1180-1182
Author(s):  
Fan Y. Chen
Keyword(s):  
Dynamic Systems ◽  
Coupled System ◽  
Circulant Matrix ◽  
Mathematical Induction ◽  
Degree Of Freedom ◽  
Recursive Solution ◽  
Direct Inspection

Based on mathematical induction, we can prove that the eigenvalues of a special nth order circulant matrix are degenerate. Hence the eigenvibration problem of an n degree-of-freedom symmetrically coupled system with equal masses and equal spring stiffnesses can be solved by direct inspection of the system schematic.

Fractal Boundary Explorations for a Nonlinear Two Degree-of-Freedom System

2012 ◽  
Author(s):  
Benjamin A. M. Owens ◽  
Brian P. Mann
Keyword(s):  
Initial Conditions ◽  
Coupled System ◽  
Basins Of Attraction ◽  
Degree Of Freedom ◽  
Fractal Boundary ◽  
Final State ◽  
Potential Wells ◽  
Mechanical Coupling ◽  
Mass Spring ◽  
Two Degree Of Freedom

This paper explores a two degree-of-freedom nonlinearly coupled system with two distinct potential wells. The system consists of a pair of linear mass-spring-dampers with a non-linear, mechanical coupling between them. This nonlinearity creates fractal boundaries for basins of attraction and forced well-escape response. The inherent uncertainty of these fractal boundaries is quantified for errors in the initial conditions and parameter space. This uncertainty relationship provides a measure of the final state and transient sensitivity of the system.

scholarly journals Numerical Integration of Multibody Dynamic Systems Involving Nonholonomic Equality Constraints

2021 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Panagiotis Passas ◽  
Elias Paraskevopoulos
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Coupled System ◽  
Weak Form ◽  
Integration Scheme ◽  
Motion Constraints ◽  
Multibody Dynamic ◽  
Time Integration Scheme

Abstract This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton’s second law and appears as a coupled system of strongly nonlinear second order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated to the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state of the art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.

Identification of Modal Damping Ratios for a Simplified Wind Turbine Tower Using Fourier Analysis

2005 ◽  
Author(s):  
Paul J. Murtagh ◽  
Biswajit Basu ◽  
Brian M. Broderick
Keyword(s):  
Linear Regression ◽  
Fourier Analysis ◽  
Wind Turbine ◽  
Order Reduction ◽  
Coupled System ◽  
Degree Of Freedom ◽  
Modal Damping ◽  
Wind Turbine Tower ◽  
Tower Model ◽  
Forced Vibration Analysis

This paper presents an approach to evaluate the modal damping ratios for a simplified wind turbine tower, using Fourier analysis and linear regression. The model proposed for the wind turbine tower is composed of a flexible tower and rotor blade system, inter-connected using a sub-structuring technique, which facilitates the rotating blade/tower coupling. A model order reduction technique is first used to model each of the two sub-structures (tower/nacelle and rotor system) as single degree-of-freedom systems. The free vibration characteristics of the tower include the effects of a large nacelle mass at the towers free end, and the corresponding properties of the rotating blades include the effects of centrifugal stiffening and axial self-weight, due to rotation. Then, the two reduced order sub systems are then coupled together to form an equivalent two degree-of-freedom coupled tower/blade wind turbine tower model. A wind-induced forced vibration analysis of the coupled tower/blades model is carried out using artificially generated wind drag time-histories obtained as discrete Fourier transform representations of wind drag power spectral density functions. From this analysis, a method is proposed, based on Fourier analysis and the linear regression, to solve the inverse problem for evaluating the first and second modal damping ratios of the coupled system. A numerical example is presented in order to demonstrate the applicability of the proposed approach, where excellent agreement was observed between the originally specified modal damping ratios and the subsequently estimated ones. The proposed method can be extended to obtain the equivalent damping of the system with soil interaction and including aerodynamic damping.

scholarly journals ‘Uncertainty’ principle in two fluid–mechanics

2020 ◽  
Vol 69 ◽  
pp. 47-55
Author(s):  
Sergey Gavrilyuk
Keyword(s):  
Potential Energy ◽  
Second Law Of Thermodynamics ◽  
Coupled System ◽  
Degree Of Freedom ◽  
Hamilton’S Principle ◽  
Governing Equations ◽  
Lagrange Equations ◽  
Hamilton's Principle ◽  
Stationary Action ◽  
Energy Equations

Hamilton’s principle (or principle of stationary action) is one of the basic modelling tools in finite-degree-of-freedom mechanics. It states that the reversible motion of mechanical systems is completely determined by the corresponding Lagrangian which is the difference between kinetic and potential energy of our system. The governing equations are the Euler-Lagrange equations for Hamil- ton’s action. Hamilton’s principle can be naturally extended to both one-velocity and multi-velocity continuum mechanics (infinite-degree-of-freedom systems). In particular, the motion of multi–velocity continuum is described by a coupled system of ‘Newton’s laws’ (Euler-Lagrange equations) for each component. The introduction of dissipative terms compatible with the second law of thermodynamics and a natural restriction on the behaviour of potential energy (convexity) allows us to derive physically reasonable and mathematically well posed governing equations. I will consider a simplest example of two-velocity fluids where one of the phases is incompressible (for example, flow of dusty air, or flow of compressible bubbles in an incompressible fluid). A very surprising fact is that one can obtain different governing equations from the same Lagrangian. Different types of the governing equations are due to the choice of independent variables and the corresponding virtual motions. Even if the total momentum and total energy equations are the same, the equations for individual components differ from each other by the presence or absence of gyroscopic forces (also called ‘lift’ forces). These forces have no influence on the hyperbolicity of the governing equations, but can drastically change the distribution of density and velocity of components. To the best of my knowledge, such an uncertainty in obtaining the governing equations of multi- phase flows has never been the subject of discussion in a ‘multi-fluid’ community.

Design of Vibratory Systems with Aid of Dimensional Analysis

1980 ◽  
Vol 102 (4) ◽  
pp. 835-841 ◽  
Author(s):  
M. S. Hundal
Keyword(s):  
Computer Program ◽  
Dimensional Analysis ◽  
Dynamic Systems ◽  
Basic Theory ◽  
Degree Of Freedom ◽  
Flexible Rotor ◽  
Vibrating System ◽  
Dimensional Parameters ◽  
Two Degree Of Freedom

The paper presents the use of dimensional analysis as an aid in the design of dynamic systems. Basic theory of dimensional analysis is discussed. Two examples are given: (1) a two degree of freedom vibrating system and (2) a flexible rotor which is flexibly supported at its ends. Choice of suitable non-dimensional parameters is discussed. A digitial computer program is described which generates all of the possible sets of non-dimensional parameters for a given problem.

scholarly journals Resonance Suppression in Multi-Degree-of-Freedom Rotating Flexible Structures Using Order-Tuned Absorbers

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Serif Gozen ◽  
Brian J. Olson ◽  
Steven W. Shaw ◽  
Christophe Pierre
Keyword(s):  
Structural Model ◽  
Flexible Structures ◽  
Coupled System ◽  
Degree Of Freedom ◽  
Tuning Parameter ◽  
Vibration Absorbers ◽  
Resonance Zone ◽  
Engine Order ◽  
Rotating Structure ◽  
Structural Mode

This paper considers the dynamic response and order-tuning of vibration absorbers fitted to a rotating flexible structure under traveling wave (TW) engine order excitation. Of specific interest is the extension of previous results on the so-called no-resonance zone, that is, a region in linear tuning parameter space in which the coupled structure/absorber system does not experience resonance over all rotation speeds. The no-resonance feature was shown to exist for cyclic rotating structures with one structural and one absorber degree of freedom (DOF) per sector. This work uses a higher-fidelity structural model to investigate the effects of higher modes on the cyclically-coupled system. It is shown that the no-resonance zone is replaced by a resonance-suppression zone in which one structural mode is suppressed, but higher-order resonances still exist with the addition of the absorbers. The results are general in the sense that one vibration mode can be eliminated using a set of identically-tuned absorbers on a rotating structure with arbitrarily many DOFs per sector.

Resonant Dynamics and Saturation in a Coupled System With Quadratic Nonlinearities

2004 ◽  
Author(s):  
Reddy Mankala ◽  
D. Dane Quinn
Keyword(s):  
Angular Velocity ◽  
Normal Modes ◽  
Coupled System ◽  
Angular Speed ◽  
Degree Of Freedom ◽  
Weakly Coupled System ◽  
Quadratic Nonlinearities ◽  
Resonant Dynamics ◽  
Three Degree Of Freedom ◽  
Dynamic Resonance

This work examines the behavior of a three-degree-of-freedom weakly coupled system. The system is composed of two components. The first is a two degree-of-freedom translational system that possesses an internal 2 : 1 resonance between the linear normal modes, which are coupled through quadratic nonlinearities. Under external forcing this component exhibits the saturation phenomena. The second is a rotational mass with a small imbalance, supported by the translational component. The angular speed of the rotor is not fixed, rather, the rotor is subject to a small torque and therefore its angular velocity slowly varies in time. A dynamic resonance occurs when the angular velocity of the rotor evolves to a neighborhood of one of the frequencies of the linear normal modes. Each of these resonances has been independently investigated previously in the literature. This work uncovers how the behavior of the dynamic resonance is modified by the mode coupling introduced by the 2 : 1 internal resonance and describes how the amplitudes of the linear normal modes are dependent on the properties of the dynamic resonance.

Harmonic Analysis of Dynamic Systems With Nonsymmetric Nonlinearities

1979 ◽  
Vol 101 (1) ◽  
pp. 31-36 ◽  
Author(s):  
P-T. D. Spanos ◽  
W. D. Iwan
Keyword(s):  
Steady State ◽  
Harmonic Analysis ◽  
Linear Systems ◽  
Dynamic Systems ◽  
Approximate Solutions ◽  
Steady State Solution ◽  
Degree Of Freedom ◽  
Equivalent Linearization ◽  
Engineering Applications ◽  
Vehicle System

The generalized method of equivalent linearization is modified to be applicable for multi-degree-of-freedom dynamic systems with nonsymmetric nonlinearities subjected to harmonic monofrequency excitation. Readily applicable formulas are given for the construction of the equivalent linear systems related to a class of systems commonly encountered in engineering applications. As an example of its application, the proposed method is used to generate an approximate steady-state solution for a simple vehicle system. The accuracy of the approximate solutions is determined.

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