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.

Periodic Motions of the Machine Tools in Cutting Process

2007 ◽  
Author(s):  
Brandon C. Gegg ◽  
Steve S. Suh ◽  
Albert C. J. Luo
Keyword(s):  
Mechanical Model ◽  
Machine Tools ◽  
Degree Of Freedom ◽  
Cutting Process ◽  
Force Distribution ◽  
Periodic Motions ◽  
Numerical Predictions ◽  
Discontinuous Boundary ◽  
Mapping Structures ◽  
Two Degree Of Freedom

In this paper, a two-degree of freedom dynamical system with discontinuity is developed to describe the vibration in the cutting process. The analytical solutions for the switchability of motion on the discontinuous boundary are presented. the switching sets based on the discontinuous boundary is introduced and the basic mappings are introduced to investigate periodic motion in such a mechanical model. The mapping structures for the stick and non-stick motions are discussed. Numerical predictions of motions of the machine tool in the cutting process are presented through the two-degree of freedom system with discontinuity. The phase trajectory and velocity and force responses are presented and the switchability of motion on the discontinuous boundary is illustrated through force distribution and force product on the boundary.

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.

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.

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.

Chaos and Quasi-Periodic Motions on the Homoclinic Surface of Nonlinear Hamiltonian Systems With Two Degrees of Freedom

2005 ◽  
Vol 1 (2) ◽  
pp. 135-142 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Energy Spectrum ◽  
Hamiltonian System ◽  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Weak Interactions ◽  
Kam Theorem ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Hamiltonian System ◽  
Two Degree Of Freedom

The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a two-degree-of-freedom (2-DOF) nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the incremental energy approach. The Poincaré mapping surfaces of chaotic motions for this specific nonlinear Hamiltonian system are illustrated. The chaotic and quasi-periodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian system. The resonant-periodic motions for such a system are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

Articulated Models of Cantilevers Conveying Fluid: The Study of a Paradox

1970 ◽  
Vol 12 (4) ◽  
pp. 288-300 ◽  
Author(s):  
M. P. Paidoussis ◽  
E. B. Deksnis
Keyword(s):  
Degrees Of Freedom ◽  
Continuous System ◽  
Dynamical Behaviour ◽  
Degree Of Freedom ◽  
Complex Frequency ◽  
Good Convergence ◽  
Articulated Models ◽  
Two Degree Of Freedom ◽  
Conveying Fluid ◽  
Oscillatory Instabilities

A general theory is presented for the dynamics of nth-degree-of-freedom articulated (lumped flexibility) models of cantilevers conveying fluid, of which the two-degree-of-freedom model of a column subjected to follower forces (first investigated by Ziegler) is a particular case. The ability of the articulated system to predict the dynamical behaviour of the continuous system modelled is investigated, and in particular the paradox that, whereas the continuous system is subject to only oscillatory instability (at sufficiently high flow), the model is generally subject to both oscillatory and buckling instabilities, and sometimes only to the latter. Complex frequency calculations show that buckling is associated with the higher modes of the articulated system, which, irrespective of the number of degrees of freedom, do not model well the corresponding modes of the continuous system. The critical flow velocities for buckling and oscillatory instabilities are calculated extensively, the latter showing good convergence to the corresponding values of the continuous system. The theory is supported by a set of experiments. Agreement between theory and experiment is satisfactorily good.

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.

Flow-Induced Vibration of Long-Span Gates

2017 ◽  
pp. 294-386
Keyword(s):  
Vortex Shedding ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Full Scale ◽  
Degree Of Freedom ◽  
Theoretical Development ◽  
Flow Induced Vibration ◽  
Long Span ◽  
Free Discharge ◽  
Two Degree Of Freedom

In this chapter the theoretical equations for fluctuating pressures due to vertical and streamwise gate motions developed in Chapters 4 and 5 are used to derive equations of motion for long-span gates with underflow, overflow and simultaneous over- and underflow. Theoretical development of analysis methods is supported by laboratory and full-scale measurements. Specifically, this chapter considers long-span gate instabilities including one degree-of-freedom vibration of gates with underflow and free discharge, one degree-of-freedom vibration of a gate with submerged discharge and vortex shedding excitation, a two degree-of-freedom vibration of long-span gates with only underflow, and two degrees-of-freedom vibration of long-span gates with simultaneous over and underflow. A method is developed to predict pressure loading on the crest of the gate with overflow.

Modeling and Experimental Evaluation of Asymmetric Pantograph Dynamics

1988 ◽  
Vol 110 (2) ◽  
pp. 168-174 ◽  
Author(s):  
S. D. Eppinger ◽  
D. N. O’Connor ◽  
W. P. Seering ◽  
D. N. Wormley
Keyword(s):  
High Performance ◽  
Degrees Of Freedom ◽  
Dynamic Models ◽  
Dynamic Testing ◽  
Dynamic Performance ◽  
Performance Model ◽  
Degree Of Freedom ◽  
Additional Degree ◽  
Three Degree Of Freedom ◽  
Two Degree Of Freedom

High-performance pantograph design requires control of pantograph dynamic performance. Many pantograph dynamic models developed to aid in the design process have employed two degrees of freedom, one for the head mass and one for the frame. In this paper, the applicability of these models to symmetric and asymmetric pantograph designs is reviewed. Two degree-of-freedom models have been shown to be appropriate to represent a number of symmetric pantograph designs. To represent the asymmetric designs considered in this paper, an additional degree of freedom representing frame dynamics has been introduced to yield a three degree-of-freedom nonlinear dynamic performance model. The model has been evaluated with experimental data obtained from laboratory dynamic testing of an asymmetric pantograph.

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