Insights on the bifurcation behavior of a freeplay system with piecewise and continuous representations.

Based on the lumped mass method, a torsional vibration model of the planetary gear system is established considering the nonlinear factors such as friction, time-varying meshing stiffness, backlash, and comprehensive error. The Runge–Kutta numerical method is used to analyze the motion characteristics of the system with various parameters and the influence of tooth friction on the bifurcation and chaos characteristics of the system. The numerical simulation results show that the system has rich bifurcation behavior with the excitation frequency, damping ratio, comprehensive error amplitude, load and backlash, and experiences multiple periodic motion and chaotic motion. Tooth friction makes the bifurcation behavior of the system fuzzy in the high frequency and heavy load areas, makes the chaos of the system restrained in the low-damping ratio and light load areas, advances the bifurcation point of the system in the small comprehensive error amplitude area, and makes the period window of the chaos area larger in the large-backlash area, which makes the bifurcation behavior of the system more complex.

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Studies of Elastic-Plastic Instabilities

Journal of Applied Mechanics ◽

10.1115/1.2789166 ◽

1999 ◽

Vol 66 (1) ◽

pp. 3-9 ◽

Cited By ~ 19

Author(s):

V. Tvergaard

Keyword(s):

Shear Band ◽

Plastic Flow ◽

Continuum Mechanics ◽

Shear Band Formation ◽

Band Formation ◽

Elastic Plastic ◽

Localized Necking ◽

Focus On Results ◽

Metal Sheets ◽

Bifurcation Behavior

Analyses of plastic instabilities are reviewed, with focus on results in structural mechanics as well as continuum mechanics. First the basic theories for bifurcation and post-bifurcation behavior are briefly presented. Then, localization of plastic flow is discussed, including shear band formation in solids, localized necking in biaxially stretched metal sheets, and the analogous phenomenon of buckling localization in structures. Also some recent results for cavitation instabilities in elastic-plastic solids are reviewed.

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Effects of some design parameters on bifurcation behavior of a magnetically supported coaxial rotor in auxiliary bearings

Engineering Computations ◽

10.1108/ec-04-2017-0141 ◽

2017 ◽

Vol 34 (7) ◽

pp. 2379-2395 ◽

Cited By ~ 2

Author(s):

Reza Ebrahimi ◽

Mostafa Ghayour ◽

Heshmatallah Mohammad Khanlo

Keyword(s):

Magnetic Bearing ◽

Active Magnetic Bearing ◽

Design Parameters ◽

Maximum Lyapunov Exponent ◽

Content Type ◽

Coaxial Rotor ◽

Rotor Bearing ◽

Auxiliary Bearing ◽

Bearing Stiffness ◽

Bifurcation Behavior

Purpose This paper aims to present bifurcation analysis of a magnetically supported coaxial rotor model in auxiliary bearings, which includes gyroscopic moments of disks and geometric coupling of the magnetic actuators. Design/methodology/approach Ten nonlinear equations of motion were solved using the Runge–Kutta method. The vibration responses were analyzed using dynamic trajectories, power spectra, Poincaré maps, bifurcation diagrams and the maximum Lyapunov exponent. The analysis was carried out for different system parameters, namely, the inner shaft stiffness, inter-rotor bearing stiffness, auxiliary bearing stiffness and disk position. Findings It was shown that dynamics of the system could be significantly affected by varying these parameters, so that the system responses displayed a rich variety of nonlinear dynamical phenomena, including quasi-periodicity, chaos and jump. Next, some threshold values were provided with regard to the design of appropriate parameters for this system. Therefore, the proposed work can provide an effective means of gaining insights into the nonlinear dynamics of coaxial rotor–active magnetic bearing systems with auxiliary bearings in the future. Originality/value This paper considered the influences of the inner shaft stiffness, inter-rotor bearing stiffness, auxiliary bearing stiffness and disk position on the bifurcation behavior of a magnetically supported coaxial rotor system in auxiliary bearings.

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Continuous representations of infinite symmetric groups on reflexive Banach spaces

Illinois Journal of Mathematics ◽

10.1215/ijm/1256051611 ◽

1973 ◽

Vol 17 (3) ◽

pp. 450-457 ◽

Cited By ~ 1

Author(s):

Arthur Lieberman

Keyword(s):

Banach Spaces ◽

Symmetric Groups ◽

Reflexive Banach Spaces ◽

Continuous Representations

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Asymmetric Bifurcation of Initially Curved Nanobeam

Journal of Applied Mechanics ◽

10.1115/1.4030647 ◽

2015 ◽

Vol 82 (9) ◽

Cited By ~ 7

Author(s):

X. Chen ◽

S. A. Meguid

Keyword(s):

Symmetry Breaking ◽

Degrees Of Freedom ◽

Surface Elasticity ◽

Beam Theory ◽

Surface Effects ◽

Beam Model ◽

Bernoulli Beam ◽

Reduced Order ◽

Euler Bernoulli Beam ◽

Bifurcation Behavior

In this paper, we investigate the asymmetric bifurcation behavior of an initially curved nanobeam accounting for Lorentz and electrostatic forces. The beam model was developed in the framework of Euler–Bernoulli beam theory, and the surface effects at the nanoscale were taken into account in the model by including the surface elasticity and the residual surface tension. Based on the Galerkin decomposition method, the model was simplified as two degrees of freedom reduced order model, from which the symmetry breaking criterion was derived. The results of our work reveal the significant surface effects on the symmetry breaking criterion for the considered nanobeam.

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