• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.13 no.4
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• pp.247-257
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• 2003

This paper presents an analytical method to Investigate the stability of a hydrodynamic journal bearing with rotating herringbone grooves. The dynamic coefficients of the hydrodynamic Journal bearing are calculated using the FEM and the perturbation method. The linear equations of motion can be represented as a parametrically excited system because the dynamic coefficients have time-varying components due to the rotating grooves, even in the steady state. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill's infinite determinant of these algebraic equations. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. This research shows that the instability of the hydrodynamic journal bearing with rotating herringbone grooves increases with increasing eccentricity and with decreasing groove number, which play the major roles in increasing the average and variation of stiffness coefficients, respectively. It also shows that a high rotational speed is another source of instability by increasing the stiffness coefficients without changing the damping coefficients.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2002.05a
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• pp.166-174
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• 2002

This paper presents an analytical method to Investigate the stability of a hydrodynamic journal bearing with rotating herringbone grooves. The dynamic coefficients of the hydrodynamic journal bearing are calculated using the FEM and the perturbation method. The linear equations of motion can be represented as a parametrically excited system because the dynamic coefficients have time-varying components due to the rotating grooves, even in the steady state. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill's infinite determinant of these algebraic equations. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. This research shows that the instability of the hydrodynamic journal bearing with rotating herringbone grooves increases with increasing eccentricity and with decreasing groove number, which play the major roles in increasing the average and variation of stiffness coefficients, respectively. It also shows that a high rotational speed is another source of instability by increasing the stiffness coefficients without changing the damping coefficients.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2002.05a
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• pp.181-189
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• 2002

This research presents an analytical model to investigate the stability due to the ball bearing waviness in a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu's equation, because the stiffness coefficients have time-varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving the Hill's infinite determinant of these algebraic equations. The validity of this research is proved by comparing the stability chart with the time responses of the vibration model suggested by prior researches. This research shows that the waviness in the rolling elements of a ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 (i= 1,2,3..).

• Coupled systems mechanics
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• v.2 no.2
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• pp.159-174
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• 2013

The present work aims at investigating the nonlinear dynamics, bifurcations, and stability of an axially accelerating beam with an intermediate spring-support. The problem of a parametrically excited system is addressed for the gyroscopic system. A geometric nonlinearity due to mid-plane stretching is considered and Hamilton's principle is employed to derive the nonlinear equation of motion. The equation is then reduced into a set of nonlinear ordinary differential equations with coupled terms via Galerkin's method. For the system in the sub-critical speed regime, the pseudo-arclength continuation technique is employed to plot the frequency-response curves. The results are presented for the system with and without a three-to-one internal resonance between the first two transverse modes. Also, the global dynamics of the system is investigated using direct time integration of the discretized equations. The mean axial speed and the amplitude of speed variations are varied as the bifurcation parameters and the bifurcation diagrams of Poincare maps are constructed.

• Journal of Advanced Marine Engineering and Technology
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• v.16 no.5
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• pp.67-76
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• 1992

This paper describes the vibration characteristic of second-order nonlinear systems subjected to parametric excitation. Emphasis is put on the examination of the hydrodynamic nonlinear damping effect on limiting the response amplitudes of parametric vibration. Since the parametric vibration is described by the Mathieu equation, the Mathieu stability chart is examined in this paper. In addition, the steady-state solutions of the nonlinear Mathieu equation in the first instability region are obtained by using a perturbation technique and are compared with those by a numerical integration method. It is shown that the response amplitudes of parametric vibration are limited even in unstable conditions by hydrodynamic nonlinear damping force. The largest reponse amplitude of parametric vibration occurs in the first instability region of Mathieu stability chart. The parametric excitation induces the response of a dynamic system to be subharmonic, superharmonic or chaotic according to their dynamic conditions.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.26 no.12
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• pp.2647-2655
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• 2002

This research presents an analytical model to investigate the stability due to the ball bearing waviness i n a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu's equation, because the stiffness coefficients have time -varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving the Hill's infinite determinant of these algebraic equations. The validity of this research is proved by comparing the stability chart with the time responses of the vibration model suggested by prior researches. This research shows that the waviness in the rolling elements of a ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 (i=1,2,3..).

• Ocean Systems Engineering
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• v.2 no.3
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• pp.229-238
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• 2012

We discuss the bi - stability that is possibly exhibited by a liquid free surface in a parametrically - driven two-dimensional (2D) rectangular tank with finite liquid depth. Following the method of adaptive mode ordering, assuming two dominant modes and retaining polynomial nonlinearities up to third-order, a nonlinear finite-dimensional nonlinear modal system approximation is obtained. A "continuation method" of nonlinear dynamics is then used in order to elicit efficiently the instability boundary in parameters' space and to predict how steady surface elevation changes as the frequency and/or the amplitude of excitation are varied. Results are compared against those of the linear version of the system (that is a Mathieu-type model) and furthermore, against an intermediate model also derived with formal mode ordering, that is based on a second - order ordinary differential equation having nonlinearities due to products of elevation with elevation velocity or acceleration. The investigation verifies that, in parameters space, there must be a region, inside the quiescent region, where liquid surface instability is exhibited. There, behaviour depends on initial conditions and a wave form would be realised only if the free surface was substantially disturbed initially.

• The Bulletin of The Korean Astronomical Society
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• v.37 no.2
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• pp.118.2-118.2
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• 2012

In particle-in-cell (PIC) simulation studies related to ion-ion two-stream instability, a reduced ion-to-electron mass ratio is often employed to save computation time. But it was not clearly verified how electrons dynamics are coupled with the slower evolution of ion-ion interactions under the external electric field. We have studied the ion beam driven instability using a 1D electrostatic PIC code by comparing different rescaling of parameter with real ion mass from the reference simulation with reduced ion mass. As the external electric field is stronger, the excited unstable mode range was more sensitively affected by the system size with the real mass ratio than the reduced ion mass. The results show that the reduced mass ratio should be used cautiously in PIC code as the electron dynamics can modify the ion instabilities. Additionally we found the formation of electron flat-top distribution in the final saturation stage. Simulation results show that in the early phase electrostatic solitary waves are quasi-periodically formed, but later they are fully dissipated resulting in heated, flat-top distributions. New electron beam components are occasionally formed. These are a consequence of the interaction with solitary wave structures. We parametrically investigate the development of electron phase space distributions for various drift speeds of ion beams and temperature ratios between ions and electrons

• Structural Engineering and Mechanics
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• v.25 no.4
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• pp.481-500
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• 2007

The dynamic instability of doubly curved panels, subjected to non-uniform tensile in-plane harmonic edge loading $P(t)=P_s+P_d\;{\cos}{\Omega}t$ is investigated. The present work deals with the problem of the occurrence of combination resonances in contrast to simple resonances in parametrically excited doubly curved panels. Analytical expressions for the instability regions are obtained at ${\Omega}={\omega}_m+{\omega}_n$, (${\Omega}$ is the excitation frequency and ${\omega}_m$ and ${\omega}_n$ are the natural frequencies of the system) by using the method of multiple scales. It is shown that, besides the principal instability region at ${\Omega}=2{\omega}_1$, where ${\omega}_1$ is the fundamental frequency, other cases of ${\Omega}={\omega}_m+{\omega}_n$, related to other modes, can be of major importance and yield a significantly enlarged instability region. The effects of edge loading, curvature, damping and the static load factor on dynamic instability behavior of simply supported doubly curved panels are studied. The results show that under localized edge loading, combination resonance zones are as important as simple resonance zones. The effects of damping show that there is a finite critical value of the dynamic load factor for each instability region below which the curved panels cannot become dynamically unstable. This example of simultaneous excitation of two modes, each oscillating steadily at its own natural frequency, may be of considerable interest in vibration testing of actual structures.