scholarly journals A Solution Procedure Combining Analytical and Numerical Approaches to Investigate a Two-Degree-of-Freedom Vibro-Impact Oscillator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1374
Author(s):  
Nicolae Herisanu ◽  
Vasile Marinca
Keyword(s):  
Analytical Solutions ◽  
Initial Conditions ◽  
Solution Procedure ◽  
Degree Of Freedom ◽  
Impact Oscillator ◽  
New Approach ◽  
Analytical Technique ◽  
The Stability ◽  
Two Degree Of Freedom ◽  
Numerical Approaches

In this paper, a new approach is proposed to analyze the behavior of a nonlinear two-degree-of-freedom vibro-impact oscillator subject to a harmonic perturbing force, based on a combination of analytical and numerical approaches. The nonlinear governing equations are analytically solved by means of a new analytical technique, namely the Optimal Auxiliary Functions Method (OAFM), which provided highly accurate explicit analytical solutions. Benefiting from these results, the application of Schur principle made it possible to analyze the stability conditions for the considered system. Various types of possible motions were emphasized, taking into account possible initial conditions and different parameters, and the explicit analytical solutions were found to be very useful to analyze the kinetic energy loss, the contact force, and the stability of periodic motions.

Bifurcation Trees of a Periodically Forced, Two-Degree-of-Freedom Oscillator With a Nonlinear Hardening Spring

2015 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
The Stability ◽  
Nonlinear Hardening ◽  
Two Degree Of Freedom

In this paper, analytical solutions of periodic motions in a periodically forced, damped, two-degree-of-freedom oscillator with a nonlinear hardening spring are obtained. The bifurcation trees of periodic motions are presented, and the stability and bifurcation of the periodic motion are determined through the eigenvalue analysis. Numerical simulations of stable period-1 and period-2 motions in the two-degree-of-freedom systems are presented, and the harmonic amplitude spectrums are presented to show the harmonic effects on periodic motions, and the accuracy of approximate analytical solutions can be estimated through the harmonic amplitudes.

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.

Galloping Vibration of Nonlinear Cables Through a Two Degree-of-Freedom Oscillator

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Transmission Line ◽  
Analytical Solutions ◽  
Transmission Lines ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Degree Of Freedom ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Two Degree Of Freedom

In this paper, galloping vibrations of a lightly iced transmission line are investigated through a two-degree-of-freedom (2-DOF) nonlinear oscillator. The 2-DOF nonlinear oscillator is used to describe the transverse and torsional motions of the galloping cables. The analytical solutions of periodic motions of galloping cables are presented through generalized harmonic balanced method. The analytical solutions of periodic motions for the galloping cable are compared with the numerical solutions, and the corresponding stability and bifurcation of periodic motions are analyzed by the eigenvalues analysis. To demonstrate the accuracy of the analytical solutions of periodic motions, the harmonic amplitudes are presented. This investigation will help one better understand galloping mechanism of iced transmission lines.

On the Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems

1959 ◽  
Vol 26 (3) ◽  
pp. 377-385
Author(s):  
R. M. Rosenberg ◽  
C. P. Atkinson
Keyword(s):  
Free Vibrations ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Natural Modes ◽  
Theoretical Predictions ◽  
Stability Chart ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

Abstract The natural modes of free vibrations of a symmetrical two-degree-of-freedom system are analyzed theoretically and experimentally. This system has two natural modes, one in-phase and the other out-of-phase. In contradistinction to the comparable single-degree-of-freedom system where the free vibrations are always orbitally stable, the natural modes of the symmetrical two-degree-of-freedom system are frequently unstable. The stability properties depend on two parameters and are easily deduced from a stability chart. For sufficiently small amplitudes both modes are, in general, stable. When the coupling spring is linear, both modes are always stable at all amplitudes. For other conditions, either mode may become unstable at certain amplitudes. In particular, if there is a single value of frequency and amplitude at which the system can vibrate in either mode, the out-of-phase mode experiences a change of stability. The experimental investigation has generally confirmed the theoretical predictions.

scholarly journals Control of Near-Grazing Dynamics in the Two-Degree-of-Freedom Vibroimpact System with Symmetrical Constraints

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zihan Wang ◽  
Jieqiong Xu ◽  
Shuai Wu ◽  
Quan Yuan
Keyword(s):  
Control Strategy ◽  
Stability Criterion ◽  
Control Method ◽  
Degree Of Freedom ◽  
Local Analysis ◽  
Grazing Bifurcation ◽  
Vibroimpact System ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.

Nonlinear Normal Modes and Localization in Elastic Vibro-Impact Systems With Multiple Constraints

2007 ◽  
Author(s):  
David Wagg
Keyword(s):  
Mathematical Formulation ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Lumped Mass ◽  
Localized Modes ◽  
Impact Oscillator ◽  
Mass System ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

In this paper we consider the dynamics of compliant mechanical systems subject to combined vibration and impact forcing. Two specific systems are considered; a two degree of freedom impact oscillator and a clamped-clamped beam. Both systems are subject to multiple motion limiting constraints. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom lumped mass system are considered. We then consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, which is a form of periodic localization. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. A numerical example of a sticking orbit for this system is shown and we discuss identifying a nonlinear normal modal basis for the system. This is achieved by extending the normal modal basis to include localized modes. Finally preliminary experimental results from a clamped-clamped vibroimpacting beam are considered and a simplified model discussed which uses an extended modal basis including localized modes.

Bifurcation Trees of Period-1 Motions to Chaos in a Two-Degree-of-Freedom, Nonlinear Oscillator

2015 ◽  
Vol 25 (13) ◽  
pp. 1550179 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Dynamical System ◽  
Fourier Series ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Series Transformation ◽  
Two Degree Of Freedom ◽  
Analytical Bifurcation Trees

In this paper, analytical solutions for period-[Formula: see text] motions in a two-degree-of-freedom (2-DOF) nonlinear oscillator are developed through the finite Fourier series. From the finite Fourier series transformation, the dynamical system of coefficients of the finite Fourier series is developed. From such a dynamical system, the solutions of period-[Formula: see text] motions are obtained and the corresponding stability and bifurcation analyses of period-[Formula: see text] motions are carried out. Analytical bifurcation trees of period-1 motions to chaos are presented. Displacements, velocities and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator.

Period-1 Motions in a Periodically Forced, Nonlinear, Machine-Tool System

2019 ◽  
Author(s):  
Siyuan Xing ◽  
Albert C. J. Luo
Keyword(s):  
Machine Tool ◽  
Numerical Simulations ◽  
Analytical Method ◽  
Degree Of Freedom ◽  
Eigenvalue Analysis ◽  
Tool Vibrations ◽  
Stability And Bifurcations ◽  
The Stability ◽  
Two Degree Of Freedom

Abstract In this paper, period-1 motions in a two-degree-of-freedom, nonlinear, machine-tool system are investigated by a semi-analytical method. The stability and bifurcations of the period-1 motions are discussed from the eigenvalue analysis. A condition is presented for the tool-and-workpiece separation in period-1 motions. Machine-tool vibrations varying with displacement disturbance from a workpiece are discussed. Numerical simulations of period-1 motions are completed from analytical predictions.

A Framework for Efficiency Evaluation of Multi-Degree-of-Freedom Gear Trains

1996 ◽  
Vol 118 (4) ◽  
pp. 556-560 ◽  
Author(s):  
C. Innocenti
Keyword(s):  
Experimental Data ◽  
Finite Number ◽  
Degree Of Freedom ◽  
Gear Train ◽  
Efficiency Evaluation ◽  
Suggested Approach ◽  
New Approach ◽  
Gear Trains ◽  
Two Degree Of Freedom

The paper proposes a new approach to the efficiency evaluation of any one- or multi-degree-of-freedom gear trains. The suggested approach generalizes the known procedures developed for two-degree-of-freedom gear trains. It is based on the determination of a vector whose components are the torques delivered to the shafts of the gear train. Furthermore the paper shows that, for a notable category of gear trains, such a vector can have only a finite number of directions, which implies that a limited number of experimental data suffices for estimating the efficiency at any operational condition. Examples of application of the proposed methodology are provided.

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