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Fulei Chu - One of the best experts on this subject based on the ideXlab platform.
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Parametric Instability of flexible rotor bearing system under time periodic base angular motions
Applied Mathematical Modelling, 2015Co-Authors: Qinkai Han, Fulei ChuAbstract:Abstract Parametric Instability of flexible rotor-bearing system under time-periodic base angular motions is analyzed in this paper. The accurate finite element model for the flexible rotor-bearing system under time-varying base angular motions is derived based upon the energy theorem and Lagrange’s principle. Three base angular motions, including the rolling, pitching and yawing motions, are assumed to be sinusoidal perturbations superimposed upon constant terms. Considering the time-varying base movements, the second order differential equations of the system will have time-periodic gyroscopic and stiffness coefficients. The discrete state transition matrix (DSTM) method is introduced for numerically acquiring the Instability regions. Based upon these, Instability computations for a rotor-bearing system with one base motion alone and two base motions together are conducted, respectively. The effects of rotating speed, amplitudes of base motion and phases between two base motions on both the primary and combination Instability regions are discussed in detail.
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Parametric Instability in Planetary Gears With Frequency-Modulated Time-Varying Mesh Stiffness
Volume 8: 26th Conference on Mechanical Vibration and Noise, 2014Co-Authors: Xinghui Qiu, Qinkai Han, Fulei ChuAbstract:A rotational model of planetary gears is developed which incorporates mesh stiffness variation and input speed fluctuations. Gear mesh stiffness is approximated by rectangle wave and different harmonic orders are considered. Because of speed fluctuations, the mesh stiffness is frequency modulated. The Parametric Instability associated with frequency-modulated time-varying stiffness is numerically investigated. The operating conditions leading to Parametric Instability are identified using Floquet theory and numerical integration. Whether the general laws derived for steady speed to suppress particular instabilities are applicable for fluctuating speed is verified. The effects of speed fluctuations on Parametric Instability are examined.Copyright © 2014 by ASME
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Parametric Instability of a jeffcott rotor with rotationally asymmetric inertia and transverse crack
Nonlinear Dynamics, 2013Co-Authors: Qinkai Han, Fulei ChuAbstract:Both the rotationally asymmetric inertia and transverse crack frequently appear in the rotor system. The Parametric excitations induced by this two features cause Instability and severe vibration under certain operating conditions. Thus, the Parametric Instability of a Jeffcott rotor with asymmetric disk and open transverse crack is studied analytically. The vibration equations of four degrees-of-freedom of the system are established, and the stiffness coefficients of cracked rotor shaft are derived based upon the compliance method and strain energy release rate method. Then, utilizing the harmonic balance method and Taylor expansion technique, the unstable widths of simple and combination Instability regions (SIR and CIR) are solved approximately. For a practical rotor system, the approximate unstable widths are verified by the Floquet numerical analysis. The effects of crack depth and position upon the unstable widths are discussed, and the conditions for zero unstable points (ZUPs) are given: Besides the asymmetric angle should be π/2 (for SIR) or 0 (for CIR), the relationships between the inertia asymmetry and crack parameters (depth and position) are also presented analytically. These results would be useful for crack detection and Instability control of the asymmetric rotor-bearing system.
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the effect of transverse crack upon Parametric Instability of a rotor bearing system with an asymmetric disk
Communications in Nonlinear Science and Numerical Simulation, 2012Co-Authors: Qinkai Han, Fulei ChuAbstract:Abstract It is well known that either the asymmetric disk or transverse crack brings Parametric inertia (or stiffness) excitation to the rotor-bearing system. When both of them appear in a rotor system, the Parametric Instability behaviors have not gained sufficient attentions. Thus, the effect of transverse crack upon Parametric Instability of a rotor-bearing system with an asymmetric disk is studied. First, the finite element equations of motion are established for the asymmetric rotor system. Both the open and breathing transverse cracks are taken into account in the model. Then, the discrete state transition matrix (DSTM) method is introduced for numerically acquiring the Instability regions. Based upon these, some computations for a practical asymmetric rotor system with open or breathing transverse crack are conducted, respectively. Variations of the primary and combination Instability regions induced by the asymmetric disk with the crack depth are observed, and the effect of the orientation angle between the crack and asymmetric disk on various Instability regions are discussed in detail. It is shown that for the asymmetric angle around 0, the existence of transverse (either open or breathing) crack has attenuation effect upon the Instability regions. Under certain crack depth, the Instability regions could be vanished by the transverse crack. When the asymmetric angle is around π/2, increasing the crack depth would enhance the Instability regions.
Qinkai Han - One of the best experts on this subject based on the ideXlab platform.
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Parametric Instability of flexible rotor bearing system under time periodic base angular motions
Applied Mathematical Modelling, 2015Co-Authors: Qinkai Han, Fulei ChuAbstract:Abstract Parametric Instability of flexible rotor-bearing system under time-periodic base angular motions is analyzed in this paper. The accurate finite element model for the flexible rotor-bearing system under time-varying base angular motions is derived based upon the energy theorem and Lagrange’s principle. Three base angular motions, including the rolling, pitching and yawing motions, are assumed to be sinusoidal perturbations superimposed upon constant terms. Considering the time-varying base movements, the second order differential equations of the system will have time-periodic gyroscopic and stiffness coefficients. The discrete state transition matrix (DSTM) method is introduced for numerically acquiring the Instability regions. Based upon these, Instability computations for a rotor-bearing system with one base motion alone and two base motions together are conducted, respectively. The effects of rotating speed, amplitudes of base motion and phases between two base motions on both the primary and combination Instability regions are discussed in detail.
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Parametric Instability in Planetary Gears With Frequency-Modulated Time-Varying Mesh Stiffness
Volume 8: 26th Conference on Mechanical Vibration and Noise, 2014Co-Authors: Xinghui Qiu, Qinkai Han, Fulei ChuAbstract:A rotational model of planetary gears is developed which incorporates mesh stiffness variation and input speed fluctuations. Gear mesh stiffness is approximated by rectangle wave and different harmonic orders are considered. Because of speed fluctuations, the mesh stiffness is frequency modulated. The Parametric Instability associated with frequency-modulated time-varying stiffness is numerically investigated. The operating conditions leading to Parametric Instability are identified using Floquet theory and numerical integration. Whether the general laws derived for steady speed to suppress particular instabilities are applicable for fluctuating speed is verified. The effects of speed fluctuations on Parametric Instability are examined.Copyright © 2014 by ASME
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Parametric Instability of a jeffcott rotor with rotationally asymmetric inertia and transverse crack
Nonlinear Dynamics, 2013Co-Authors: Qinkai Han, Fulei ChuAbstract:Both the rotationally asymmetric inertia and transverse crack frequently appear in the rotor system. The Parametric excitations induced by this two features cause Instability and severe vibration under certain operating conditions. Thus, the Parametric Instability of a Jeffcott rotor with asymmetric disk and open transverse crack is studied analytically. The vibration equations of four degrees-of-freedom of the system are established, and the stiffness coefficients of cracked rotor shaft are derived based upon the compliance method and strain energy release rate method. Then, utilizing the harmonic balance method and Taylor expansion technique, the unstable widths of simple and combination Instability regions (SIR and CIR) are solved approximately. For a practical rotor system, the approximate unstable widths are verified by the Floquet numerical analysis. The effects of crack depth and position upon the unstable widths are discussed, and the conditions for zero unstable points (ZUPs) are given: Besides the asymmetric angle should be π/2 (for SIR) or 0 (for CIR), the relationships between the inertia asymmetry and crack parameters (depth and position) are also presented analytically. These results would be useful for crack detection and Instability control of the asymmetric rotor-bearing system.
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the effect of transverse crack upon Parametric Instability of a rotor bearing system with an asymmetric disk
Communications in Nonlinear Science and Numerical Simulation, 2012Co-Authors: Qinkai Han, Fulei ChuAbstract:Abstract It is well known that either the asymmetric disk or transverse crack brings Parametric inertia (or stiffness) excitation to the rotor-bearing system. When both of them appear in a rotor system, the Parametric Instability behaviors have not gained sufficient attentions. Thus, the effect of transverse crack upon Parametric Instability of a rotor-bearing system with an asymmetric disk is studied. First, the finite element equations of motion are established for the asymmetric rotor system. Both the open and breathing transverse cracks are taken into account in the model. Then, the discrete state transition matrix (DSTM) method is introduced for numerically acquiring the Instability regions. Based upon these, some computations for a practical asymmetric rotor system with open or breathing transverse crack are conducted, respectively. Variations of the primary and combination Instability regions induced by the asymmetric disk with the crack depth are observed, and the effect of the orientation angle between the crack and asymmetric disk on various Instability regions are discussed in detail. It is shown that for the asymmetric angle around 0, the existence of transverse (either open or breathing) crack has attenuation effect upon the Instability regions. Under certain crack depth, the Instability regions could be vanished by the transverse crack. When the asymmetric angle is around π/2, increasing the crack depth would enhance the Instability regions.
Peter Goldreich - One of the best experts on this subject based on the ideXlab platform.
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gravity modes in zz ceti stars iv amplitude saturation by Parametric Instability
The Astrophysical Journal, 2001Co-Authors: Peter GoldreichAbstract:ZZ Ceti stars (also known as DAV stars) exhibit small-amplitude photometric pulsations in multiple gravity modes. As the stars cool, their dominant modes shift to longer periods. We demonstrate that Parametric Instability limits overstable modes to amplitudes similar to those observed. In particular, it reproduces the trend that longer period modes have larger amplitudes. Parametric Instability is a form of resonant three-mode coupling. It involves the destabilization of a pair of stable daughter modes by an overstable parent mode. The three modes must satisfy exact angular selection rules and approximate frequency resonance. The lowest Instability threshold for each parent mode is provided by the daughter pair that minimizes (δω^2 + γ^2_d)/κ^2, where κ is the nonlinear coupling constant, δω is the frequency mismatch, and γ_d is the energy damping rate of the daughter modes. Parametric Instability leads to a steady state if |δω| > γ_d and to limit cycles if |δω| < γ_d. The former behavior characterizes low radial order (n ≤ 3) parent modes, and the latter those with n ≥ 5. In either case, the overstable mode's amplitude is maintained at close to the Instability threshold value. Although Parametric Instability defines an upper envelope for the amplitudes of overstable modes in ZZ Ceti stars, other nonlinear mechanisms are required to account for the irregular distribution of amplitudes of similar modes and the nondetection of modes with periods longer than 1200 s. Resonant three-mode interactions involving more than one excited mode may account for the former and Kelvin-Helmholtz Instability of the mode-driven shear layer below the convection zone for the latter.
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gravity modes in zz ceti stars iv amplitude saturation by Parametric Instability
arXiv: Astrophysics, 2000Co-Authors: Peter GoldreichAbstract:ZZ Ceti stars exhibit small amplitude photometric pulsations in multiple gravity-modes. We demonstrate that Parametric Instability, a form of resonant 3-mode coupling, limits overstable modes to amplitudes similar to those observed. In particular, it reproduces the observed trend that longer period modes have larger amplitudes. Parametric Instability involves the destabilization of a pair of stable daughter modes by an overstable parent mode. The 3-modes must satisfy exact angular selection rules and approximate frequency resonance. The lowest Instability threshold for each parent mode is provided by the daughter pair that minimizes $(\delta\omega^2+\gamma_d^2)/\kappa^2$, where $\kappa$ is the nonlinear coupling constant, $\delta\omega$ is the frequency mismatch, and $\gamma_d$ is the energy damping rate of the daughter modes. The overstable mode's amplitude is maintained at close to the Instability threshold value. Although Parametric Instability defines an upper envelope for the amplitudes of overstable modes in ZZ Ceti stars, other nonlinear mechanisms are required to account for the irregular distribution of amplitudes of similar modes and the non-detection of modes with periods longer than $1,200\s$. Resonant 3-mode interactions involving more than one excited mode may account for the former. Our leading candidate for the latter is Kelvin-Helmholtz Instability of the mode-driven shear layer below the convection zone.
Hamed Farokhi - One of the best experts on this subject based on the ideXlab platform.
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Parametric Instability of microbeams in supercritical regime
Nonlinear Dynamics, 2016Co-Authors: Mergen H. Ghayesh, Hamed FarokhiAbstract:The supercritical Parametric Instability of a microbeam subject to a time-dependent axial load is examined. The axial load is comprised of a constant mean value along with harmonic fluctuations. The mean value is increased from zero and is set to a value in the supercritical regime; the nonlinear Parametric Instability over the buckled state, due to the axial load variations, is examined. From the modelling perspective, based on the modified couple stress theory, the potential energy of the system is obtained in terms of the system parameters and displacement field. Moreover, the kinetic energy is formulated as a function of system parameters and the displacement field. The continuous model is developed by means of Hamilton’s principle and then truncated into a reduced-order model via a weighted-residual technique. Three different numerical techniques, i.e. the pseudo-arclength continuation method, a direct time-integration scheme, and an eigenvalue extraction, are employed to solve the high-dimensional reduced-order model. A stability analysis is also conducted via the Floquet theory. The nonlinear size-dependent Parametric response of the system over the buckled configuration is presented in the form of frequency–response diagrams, force–response curves, time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.
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Parametric Instability of microbeams in supercritical regime
Nonlinear Dynamics, 2015Co-Authors: Mergen H. Ghayesh, Hamed FarokhiAbstract:The supercritical Parametric Instability of a microbeam subject to a time-dependent axial load is examined. The axial load is comprised of a constant mean value along with harmonic fluctuations. The mean value is increased from zero and is set to a value in the supercritical regime; the nonlinear Parametric Instability over the buckled state, due to the axial load variations, is examined. From the modelling perspective, based on the modified couple stress theory, the potential energy of the system is obtained in terms of the system parameters and displacement field. Moreover, the kinetic energy is formulated as a function of system parameters and the displacement field. The continuous model is developed by means of Hamilton’s principle and then truncated into a reduced-order model via a weighted-residual technique. Three different numerical techniques, i.e. the pseudo-arclength continuation method, a direct time-integration scheme, and an eigenvalue extraction, are employed to solve the high-dimensional reduced-order model. A stability analysis is also conducted via the Floquet theory. The nonlinear size-dependent Parametric response of the system over the buckled configuration is presented in the form of frequency–response diagrams, force–response curves, time histories, phase-plane portraits, fast Fourier transforms, and Poincare sections.
Mergen H. Ghayesh - One of the best experts on this subject based on the ideXlab platform.
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Parametric Instability of microbeams in supercritical regime
Nonlinear Dynamics, 2016Co-Authors: Mergen H. Ghayesh, Hamed FarokhiAbstract:The supercritical Parametric Instability of a microbeam subject to a time-dependent axial load is examined. The axial load is comprised of a constant mean value along with harmonic fluctuations. The mean value is increased from zero and is set to a value in the supercritical regime; the nonlinear Parametric Instability over the buckled state, due to the axial load variations, is examined. From the modelling perspective, based on the modified couple stress theory, the potential energy of the system is obtained in terms of the system parameters and displacement field. Moreover, the kinetic energy is formulated as a function of system parameters and the displacement field. The continuous model is developed by means of Hamilton’s principle and then truncated into a reduced-order model via a weighted-residual technique. Three different numerical techniques, i.e. the pseudo-arclength continuation method, a direct time-integration scheme, and an eigenvalue extraction, are employed to solve the high-dimensional reduced-order model. A stability analysis is also conducted via the Floquet theory. The nonlinear size-dependent Parametric response of the system over the buckled configuration is presented in the form of frequency–response diagrams, force–response curves, time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.
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Parametric Instability of microbeams in supercritical regime
Nonlinear Dynamics, 2015Co-Authors: Mergen H. Ghayesh, Hamed FarokhiAbstract:The supercritical Parametric Instability of a microbeam subject to a time-dependent axial load is examined. The axial load is comprised of a constant mean value along with harmonic fluctuations. The mean value is increased from zero and is set to a value in the supercritical regime; the nonlinear Parametric Instability over the buckled state, due to the axial load variations, is examined. From the modelling perspective, based on the modified couple stress theory, the potential energy of the system is obtained in terms of the system parameters and displacement field. Moreover, the kinetic energy is formulated as a function of system parameters and the displacement field. The continuous model is developed by means of Hamilton’s principle and then truncated into a reduced-order model via a weighted-residual technique. Three different numerical techniques, i.e. the pseudo-arclength continuation method, a direct time-integration scheme, and an eigenvalue extraction, are employed to solve the high-dimensional reduced-order model. A stability analysis is also conducted via the Floquet theory. The nonlinear size-dependent Parametric response of the system over the buckled configuration is presented in the form of frequency–response diagrams, force–response curves, time histories, phase-plane portraits, fast Fourier transforms, and Poincare sections.