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.

Period-3 Motions in a Two Degree-of-Freedom Nonlinear Oscillator

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Degrees Of Freedom ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Amplitude Spectra ◽  
Stability And Bifurcations ◽  
Two Degree Of Freedom

In this paper, periodic motions of a two-degree-of-freedom nonlinear oscillator are studied by using general harmonic balanced method. Stable and unstable period-3 motions are obtained. The corresponding stability and bifurcations of the period-3 motions are determined through the eigenvalue analysis. Both symmetric and asymmetric period-3 motions are found in the system with a certain set of parameter. Numerical simulations of both stable and unstable period-3 motions in the two degrees of freedom systems are illustrated. The harmonic amplitude spectra show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be observed.

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.

Period-1 Motions in a Two-Degree-of-Freedom Nonlinear Oscillator With Periodic Excitation

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

In this paper, analytical solutions for period-1 motions in a periodically forced, two-degrees-of-freedom system with a nonlinear spring are developed. The stability and bifurcation of the periodic motions are completed by the eigenvalue analysis. Both symmetric and asymmetric periodic motions are found in the system. Analytical solutions of both stable and unstable period-1 are presented. Finally, numerical simulations of stable and unstable motions in the two degrees of freedom systems are presented. The harmonic amplitude spectrums show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be observed.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

2014 ◽  
Vol 24 (05) ◽  
pp. 1450075 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Fourier Series ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Series Solution ◽  
Parametric Oscillator ◽  
Periodic Motions ◽  
Parametric Oscillators ◽  
Harmonic Term ◽  
Nonlinear Behaviors

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

D11 Elliptic Averaging Method for Nonlinear Oscillator Having Two-Degree-of-Freedom

2011 ◽  
Vol 2011 (0) ◽  
pp. 97-98
Author(s):  
Tadashi OKABE ◽  
Takahiro KONDOU ◽  
Jumpei OHNISHI ◽  
Hideo TORIHARA
Keyword(s):  
Nonlinear Oscillator ◽  
Averaging Method ◽  
Degree Of Freedom ◽  
Elliptic Averaging ◽  
Two Degree Of Freedom

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.

Modified Homotopy Analysis Method for Nonlinear Aeroelastic Behavior of Two Degree-of-Freedom Airfoils

2016 ◽  
Vol 16 (09) ◽  
pp. 1520001 ◽  
Author(s):  
Yaobin Niu ◽  
Zhongwei Wang ◽  
Dequan Wang ◽  
Bing Liu
Keyword(s):  
Homotopy Analysis Method ◽  
Control Parameter ◽  
Numerical Solutions ◽  
Degree Of Freedom ◽  
Aerodynamic Load ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
Two Degree Of Freedom ◽  
Convergence Control Parameter

In this paper, the homotopy analysis method (HAM) is extended to deal with the nonlinear aeroelastic problem of a two degree-of-freedom (DOF) airfoil. To avoid determination of the parameter for the complicated high-order minimization problem, a new modified HAM is proposed based on the idea of minimizing the squared residual. Using this method, the convergence-control parameter is determined by the low order squared residual of the governing equations, and then the problem is solved in a way similar to the basic HAM. The proposed method is used to solve the nonlinear aeroelastic behavior of a supersonic airfoil, with the unsteady aerodynamic load evaluated by the piston theory. Two examples are prepared, for which the frequencies and amplitudes of the limit cycles are obtained. The approximate solutions obtained are demonstrated to agree excellently the numerical solutions, meanwhile, the convergence-control parameter can be easily determined using the present approach.

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