Free Periodic Motions of an Undamped Two-Degree-of-Freedom Oscillatory System With Nonlinear Unsymmetrical Elasticity

1950 ◽  
Vol 17 (2) ◽  
pp. 185-190
Author(s):  
Walter W. Soroka
Keyword(s):  
Nonlinear Systems ◽  
Linear Systems ◽  
Oscillatory System ◽  
Degree Of Freedom ◽  
The Other ◽  
Periodic Motions ◽  
Deflection Curve ◽  
Approximate Treatment ◽  
Load Deflection Curve ◽  
Two Degree Of Freedom

Abstract Precise solutions are presented for a two-degree-of-freedom oscillatory system containing a preset spring. Such a system is characteristic of an aircraft propeller-engine-supercharger installation. The periodic free motions obtained indicate the possibility of highly unconventional motions when the nonlinearity is pronounced, motions which may be easily overlooked in the usual approximate treatment of nonlinear systems. The results presented in this paper show that one mass may oscillate several times while the other mass is going through one oscillation. The ratio of oscillations of one mass with respect to the other changes with amplitude. By considering the load-deflection curve for the nonlinear spring to be a broken line, periodic motions may be obtained to any desired degree of precision by combining conventional general solutions for two-degree-of-freedom linear systems.

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.

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.

scholarly journals Snapping of a Planar Elastica With Fixed End Slopes

2008 ◽  
Vol 75 (4) ◽  
Author(s):  
Jen-San Chen ◽  
Yong-Zhi Lin
Keyword(s):  
Theoretical Prediction ◽  
Static Load ◽  
Natural Frequencies ◽  
Equilibrium Configuration ◽  
The Other ◽  
Deflection Curve ◽  
Slope Difference ◽  
The Stability ◽  
Fixed End ◽  
Load Deflection Curve

In this paper, we study the deformation and stability of a planar elastica. One end of the elastica is clamped and fixed in space. The other end of the elastica is also clamped, but the clamp itself is allowed to slide along a linear track with a slope different from that of the fixed clamp. The elastica deforms after it is subjected to an external pushing force on the moving clamp. It is observed that when the pushing force reaches a critical value, snapping may occur as the elastica jumps from one configuration to another remotely away from the original one. In the theoretical investigation, we calculate the static load-deflection curve for a specified slope difference between the fixed clamp and the moving clamp. To study the stability of the equilibrium configuration, we superpose the equilibrium configuration with a small perturbation and calculate the natural frequencies of the deformed elastica. An experimental setup is designed to measure the load-deflection curve and the natural frequencies of the elastica. The measured load-deflection relation agrees with the theoretical prediction very well. On the other hand, the measured natural frequencies do not agree very well with the theoretical prediction, unless the mass of the moving clamp is taken into account.

scholarly journals Pendekatan Eksperimen Karakteristik Respon Getaran Sistem Two Degree of Freedom dengan Penambahan Independent Dual Dynamic Vibration Absorber

2019 ◽  
Vol 3 (2) ◽  
pp. 85
Author(s):  
Susastro Susastro ◽  
Novi Indah Riani
Keyword(s):  
Vibration Reduction ◽  
Degree Of Freedom ◽  
The Other ◽  
Vibration Absorber ◽  
Dynamic Vibration Absorber ◽  
Maximum Reduction ◽  
Dynamic Vibration ◽  
Translational Vibration ◽  
Mathematical Equations ◽  
Two Degree Of Freedom

Vibration is one of the problems that must be reduced in a vehicle. There are many ways to reduce vibration in vehicles, one of them is by adding Dynamic vibration absorber (DVA). While Dual Dynamic vibration absorber (dDVA) is a DVA period that is able to move in the translational direction given to the system to reduce translation vibration and when there is resonance. Translation DVA is an additional type of time used to reduce the vibration of the translation direction. So far there is not much research related to the use of translational DVA to reduce rotational vibrations as well as translation. In this study, a study was conducted related to the use of independent double translational DVA (dDVA) to reduce translation vibrations as well as rotation of the beam. The research was conducted by modeling the system obtained into mathematical equations and simulations were carried out to determine the characteristics of vibrations that arise. In the simulation, one of the DVA periods is placed at the center of the main system period, while the other DVA period is given a change between the center period and the end of the system. The results of the study show that the maximum reduction in translational vibration is 95.51% and occurs when the absorber is placed at the center of the system, while the maximum rotation vibration reduction is 56.62% and is obtained when the system is given with an arm ratio of 1 and zero.

On Motion Switchability in a Two Degree of Freedom, Friction-Induced Oscillator Traveling on Constant Speed Belts

2009 ◽  
Author(s):  
Albert C. J. Luo ◽  
Tingting Mao
Keyword(s):  
Degrees Of Freedom ◽  
Degree Of Freedom ◽  
Constant Speed ◽  
Periodic Motions ◽  
Mapping Structures ◽  
Two Degree Of Freedom ◽  
Motion Complexity

In this paper, all possible stick and non-stick motions in such a friction-induced oscillator are discussed and the corresponding analytical conditions for the stick and non-stick motions to the traveling belts are presented. The mapping structures are introduced and the periodic motions of the two oscillators are presented through the corresponding mapping structure. Velocity and force responses for stick and non-stick, periodic motions in the 2-DOF friction-induced system are illustrated for a better understanding of the motion complexity in such many degrees of freedom systems.

Vibration Response of Linear Damped Complex Systems

1963 ◽  
Vol 30 (1) ◽  
pp. 70-74 ◽  
Author(s):  
Robert Plunkett
Keyword(s):  
Linear System ◽  
Complex Systems ◽  
Normal Modes ◽  
Continuous System ◽  
Vibration Response ◽  
Degree Of Freedom ◽  
The Other ◽  
Maximum Response ◽  
Approximate Expressions ◽  
Two Degree Of Freedom

Hahnkamm has found the changes in the amplitudes of each of the two maxima of the unit vibration response of a two-degree-of-freedom linear system as the strength of the single linear dashpot is changed. This paper develops two approximate expressions for the change in all of the response maxima of a multidegree or continuous system as the dashpot constant of the single linear damper is changed. One of these approximations is derived from a perturbation solution around the minimax values, and the other is derived from an expansion in normal modes. These expressions are useful in determining the sensitivity of the maximum response value to small changes in the damping constant.

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