scholarly journals Chaotic and Subharmonic Motion Analysis of Floating Ring Gas Bearing System by Hybrid Numerical Method

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Cheng-Chi Wang ◽  
Her-Terng Yau ◽  
Chi-Chang Wang
Keyword(s):  
Finite Difference ◽  
Numerical Method ◽  
Nonlinear Dynamic ◽  
Computational Time ◽  
Time Step ◽  
Hybrid Numerical Method ◽  
Bearing Number ◽  
Gas Bearing ◽  
Rotor Mass ◽  
Bearing System

This paper studies the nonlinear dynamic behaviors including chaotic, subharmonic, and quasi-periodic motions of a rigid rotor supported by floating ring gas bearing (FRGB) system. A hybrid numerical method combining the differential transformation method and the finite difference method used to calculate pressure distribution of FRGB system and rotor orbits. The results obtained for the orbits of the rotor center are in good agreement with those obtained using the traditional finite difference approach. Moreover, the hybrid method avoids the numerical instability problem suffered by the finite difference scheme at low values of the rotor mass and computational time-step. Moreover, power spectra, Poincaré maps, bifurcation diagrams and Lyapunov exponents are applied to examine the nonlinear dynamic response of the FRGB system over representative ranges of the rotor mass and bearing number, respectively. The results presented summarize the changes which take place in the dynamic behavior of the FRGB system as the rotor mass and bearing number are increased and therefore provide a useful guideline for the bearing system.

scholarly journals Bifurcation and Nonlinear Dynamic Analysis of Externally Pressurized Double Air Films Bearing System

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Cheng-Chi Wang ◽  
Chin-Chia Liu ◽  
Chi-Chang Wang
Keyword(s):  
Finite Difference ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic Analysis ◽  
Transformation Method ◽  
Bifurcation Phenomenon ◽  
System A ◽  
Hybrid Numerical Method ◽  
Rotor Center ◽  
Bearing Number ◽  
Bearing System

This paper studies the chaotic and nonlinear dynamic behaviors of a rigid rotor supported by externally pressurized double air films (EPDAF) bearing system. A hybrid numerical method combining the differential transformation method and the finite difference method is used to calculate pressure distribution of EPDAF bearing system and bifurcation phenomenon of rotor center orbits. The results obtained for the orbits of the rotor center are in good agreement with those obtained using the traditional finite difference approach. The results presented summarize the changes which take place in the dynamic behavior of the EPDAF bearing system as the rotor mass and bearing number are increased and therefore provide a useful guideline for the bearing system.

FINITE DIFFERENCE METHOD FOR SOLVING THE NONLINEAR DYNAMIC EQUATION OF UNDERWATER TOWED SYSTEM

2014 ◽  
Vol 11 (04) ◽  
pp. 1350060 ◽  
Author(s):  
ZHIJIANG YUAN ◽  
LIANGAN JIN ◽  
WEI CHI ◽  
HENGDOU TIAN
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Difference Method ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Computational Time ◽  
Algebraic Equations ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Dynamic Equations

A wide body of work exists that describes numerical solution for the nonlinear system of underwater towed system. Many researchers usually divide the tow cable with less number elements for the consideration of computational time. However, this type of installation affects the accuracy of the numerical solution. In this paper, a newly finite difference method for solving the nonlinear dynamic equations of the towed system is developed. The mathematical model of tow cable and towed body are both discretized to nonlinear algebraic equations by center finite difference method. A newly discipline for formulating the nonlinear equations and Jacobian matrix of towed system are proposed. We can solve the nonlinear dynamic equation of underwater towed system quickly by using this discipline, when the size of number elements is large.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T175-T193 ◽  
Author(s):  
Enjiang Wang ◽  
Jing Ba ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Computational Time ◽  
Time Step ◽  
Space Domain ◽  
Time Space ◽  
Temporal Dispersion

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.

Nonlinear Dynamic Analysis of a Flexible Rotor Supported by Externally Pressurized Porous Gas Journal Bearings

2002 ◽  
Vol 124 (3) ◽  
pp. 553-561 ◽  
Author(s):  
Cheng-Chi Wang ◽  
Cheng-Ying Lo ◽  
Cha’o-Kuang Chen
Keyword(s):  
Dynamic Analysis ◽  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic Analysis ◽  
Power Spectra ◽  
Journal Bearings ◽  
Operating Conditions ◽  
Flexible Rotor ◽  
Bearing Number ◽  
Rotor Mass

This paper studies the nonlinear dynamic analysis of a flexible rotor supported by externally pressurized porous gas journal bearings. A time-dependent mathematical model for externally pressurized porous gas journal bearings is presented. The finite difference method and the Successive Over Relation (S.O.R.) method are employed to solve the modified Reynolds’ equation. The system state trajectory, Poincare´ maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and journal center in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and quasi-periodic response of the rotor and journal center. This paper shows how the dynamic behavior of this type of system varies with changes in rotor mass and bearing number. The results of this study contribute to a further understanding of the nonlinear dynamics of gas-lubricated, externally pressurized, porous rotor-bearing systems.

Chaos and Nonlinear Dynamic Analysis of Porous Air Bearing System

2015 ◽  
Vol 764-765 ◽  
pp. 204-207
Author(s):  
Cheng Chi Wang ◽  
Jui Pin Hung
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Effective Means ◽  
Nonlinear Dynamic Analysis ◽  
Transformation Method ◽  
Dynamic Responses ◽  
Air Bearing ◽  
Dynamic Behaviors ◽  
Rotor Mass ◽  
Bearing System

The chaos and nonlinear dynamic behaviors of porous air bearing system are studied by a hybrid numerical method combining the finite difference method (FDM) and differential transformation method (DTM). The numerical results are verified by two different schemes including hybrid method and FDM and the current analytical results are found to be in good agreement. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass is increased. From the dynamic responses of the rotor center, they reveal complex dynamic behaviors including periodic, sub-harmonic motion and chaos. The results of this study provide an understanding of the nonlinear dynamic behavior of PAB systems characterized by different rotor masses. Specifically, the results have shown that system exists chaotic motion over the ranges of rotor mass 10.66≤Mr<13.7kg. The proposed method and results provide an effective means of gaining insights into the porous air bearing systems.

scholarly journals Application of Adjoint Data Assimilation Method to Atmospheric Aerosol Transport Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Minjie Xu ◽  
Kai Fu ◽  
Xianqing Lv
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Large Scale ◽  
Large Time ◽  
Computational Time ◽  
Aerosol Transport ◽  
Time Step ◽  
Adjoint Assimilation ◽  
Transport Problems

We propose combining the adjoint assimilation method with characteristic finite difference scheme (CFD) to solve the aerosol transport problems, which can predict the distribution of atmospheric aerosols efficiently by using large time steps. Firstly, the characteristic finite difference scheme (CFD) is tested to compute the Gaussian hump using large time step sizes and is compared with the first-order upwind scheme (US1) using small time steps; the US1 method gets E2 error of 0.2887 using Δt=1/450, while CFD method gets a much smaller E2 of 0.2280 using a much larger time step Δt=1/45. Then, the initial distribution of PM2.5 concentration is inverted by the adjoint assimilation method with CFD and US1. The adjoint assimilation method with CFD gets better accuracy than adjoint assimilation method with US1 while adjoint assimilation method with CFD costs much less computational time. Further, a real case of PM2.5 concentration distribution in China during the APEC 2014 is simulated by using adjoint assimilation method with CFD. The simulation results are in good agreement with the observed values. The adjoint assimilation method with CFD can solve large scale aerosol transport problem efficiently.

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