scholarly journals The Transient POD Method Based on Minimum Error of Bifurcation Parameter

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 392
Author(s):  
Kuan Lu ◽  
Haopeng Zhang ◽  
Kangyu Zhang ◽  
Yulin Jin ◽  
Shibo Zhao ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Order Reduction ◽  
Optimal Sampling ◽  
Bifurcation Parameter ◽  
Sliding Bearings ◽  
Minimum Error ◽  
Reduction Model ◽  
Model Equivalence ◽  
Proper Orthogonal

An invariable order reduction model cannot be obtained by the adaptive proper orthogonal decomposition (POD) method in parametric domain, there exists uniqueness of the model with different conditions. In this paper, the transient POD method based on the minimum error of bifurcation parameter is proposed and the order reduction conditions in the parametric domain are provided. The order reduction model equivalence of optimal sampling length is discussed. The POD method was applied for order reduction of a high-dimensional rotor system supported by sliding bearings in a certain speed range. The effects of speed, initial conditions, sampling length, and mode number on parametric domain order reduction are discussed. The existence of sampling length was verified, and two- and three-degrees-of-freedom (DOF) invariable order reduction models were obtained by proper orthogonal modes (POM) on the basis of optimal sampling length.

scholarly journals Dynamical Behaviors Analysis of the Rotor Model with Coupling Faults and Applications of the TPOD Method

2020 ◽  
Vol 10 (21) ◽  
pp. 7415
Author(s):  
Kuan Lu ◽  
Nan Wu ◽  
Kangyu Zhang ◽  
Chao Fu ◽  
Yulin Jin ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Order Reduction ◽  
Rotor System ◽  
Optimal Order ◽  
Sliding Bearings ◽  
Reduction Model ◽  
Coupling Faults ◽  
Order Reduction Method ◽  
Proper Orthogonal ◽  
Rotor Model

The transient proper orthogonal decomposition (TPOD) method is applied for order reduction in the rotor-bearing system with the coupling faults in this paper. A 24 degrees of freedom (DOFs) rotor model supported by a pair of sliding bearings with both crack and rub-impact faults is established by the discrete modeling method. The complexity of dynamic behaviors of the rotor system with the coupling faults is discussed via the comparison of the rotor system with the single fault (crack or rub-impact). The proper orthogonal mode (POM) energy method is proposed to confirm the DOF number of the reduced model. The TPOD method is used in the coupling faults system to obtain the optimal order reduction model based on the POM energy. The efficiency of the order reduction method is verified by comparing the bifurcation behaviors between the original and the reduced system. The TPOD method provides the optimal order reduction model to study the non-linear dynamic characteristics of the complex rotor system with the coupling faults.

A Dual-Weighted Approach to Order Reduction in 4DVAR Data Assimilation

2008 ◽  
Vol 136 (3) ◽  
pp. 1026-1041 ◽  
Author(s):  
D. N. Daescu ◽  
I. M. Navon
Keyword(s):  
Data Assimilation ◽  
Proper Orthogonal Decomposition ◽  
Initial Conditions ◽  
Order Reduction ◽  
Orthogonal Decomposition ◽  
Low Rank ◽  
Water Model ◽  
Reduced Order ◽  
Model Dynamics ◽  
Proper Orthogonal

Abstract Strategies to achieve order reduction in four-dimensional variational data assimilation (4DVAR) search for an optimal low-rank state subspace for the analysis update. A common feature of the reduction methods proposed in atmospheric and oceanographic studies is that the identification of the basis functions relies on the model dynamics only, without properly accounting for the specific details of the data assimilation system (DAS). In this study a general framework of the proper orthogonal decomposition (POD) method is considered and a cost-effective approach is proposed to incorporate DAS information into the order-reduction procedure. The sensitivities of the cost functional in 4DVAR data assimilation with respect to the time-varying model state are obtained from a backward integration of the adjoint model. This information is further used to define appropriate weights and to implement a dual-weighted proper orthogonal decomposition (DWPOD) method for order reduction. The use of a weighted ensemble data mean and weighted snapshots using the adjoint DAS is a novel element in reduced-order 4DVAR data assimilation. Numerical results are presented with a global shallow-water model based on the Lin–Rood flux-form semi-Lagrangian scheme. A simplified 4DVAR DAS is considered in the twin-experiment framework with initial conditions specified from the 40-yr ECMWF Re-Analysis (ERA-40) datasets. A comparative analysis with the standard POD method shows that the reduced DWPOD basis may provide an increased efficiency in representing an a priori specified forecast aspect and as a tool to perform reduced-order optimal control. This approach represents a first step toward the development of an order-reduction methodology that combines in an optimal fashion the model dynamics and the characteristics of the 4DVAR DAS.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1995 ◽  
Vol 117 (3) ◽  
pp. 582-588 ◽  
Author(s):  
L. N. Virgin ◽  
T. F. Walsh ◽  
J. D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations, which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions, and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors, which described the practical context of the work, free vibration, control aspects, and derivation of the mathematical model.

A Quasi-three-dimensional Finite-volume Shallow Water Model for Green Water on Deck

2013 ◽  
Vol 57 (03) ◽  
pp. 125-140
Author(s):  
Daniel A. Liut ◽  
Kenneth M. Weems ◽  
Tin-Guen Yen
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Time Domain ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Green Water ◽  
Water Model ◽  
Ship Motions ◽  
Shallow Water Model

A quasi-three-dimensional hydrodynamic model is presented to simulate shallow water phenomena. The method is based on a finite-volume approach designed to solve shallow water equations in the time domain. The nonlinearities of the governing equations are considered. The methodology can be used to compute green water effects on a variety of platforms with six-degrees-of-freedom motions. Different boundary and initial conditions can be applied for multiple types of moving platforms, like a ship's deck, tanks, etc. Comparisons with experimental data are discussed. The shallow water model has been integrated with the Large Amplitude Motions Program to compute the effects of green water flow over decks within a time-domain simulation of ship motions in waves. Results associated to this implementation are presented.

Application of Simplified Parametric Model to Estimate Fan Blade-Out Response

2018 ◽  
Author(s):  
Theodore S. Brockett ◽  
Jerzy T. Sawicki
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Engine Performance ◽  
Low Pressure ◽  
External Loading ◽  
System Response ◽  
Angular Rotation ◽  
Mass Eccentricity ◽  
Fan Blade ◽  
The Impact

A six-degree-of-freedom non-linear model is developed using Lagrange’s equation. The model is used to estimate transient fan-stage dynamic response during a fan-blade-out event in a turbo fan engine. The coupled degrees of freedom in the model include the fan whirl in the fan plane, the torsional response of the fan and low-pressure turbines (LPTs) about the engine centerline, the radial position of the released blade fragment, and the angular rotation of the trailing blade from its free state due to acceleration of the released blade. The released blade is assumed to slide radially outward along the trailing blade without friction. The external loading applied to the system includes fan imbalance, the remaining fan blades machining away the rub strip, rubbing of the blades with the fan case, and slowly-varying torques on the low pressure (LP) spool as engine performance degrades. The machining of the abradable imparts tangential loading on the fan blades as momentum is transferred to the liberated rub strip material. After application of the initial conditions including angular positions, angular velocities, released blade fragment position, and torsional wind-up, the governing equations are integrated forward in time from the instant the blade fragment is released. A reasonable match to test data is shown. Parameters affecting the fan-system response are varied to study the impact on fan peak lateral whirl amplitude, peak LP shaft torque, and peak loading on the trailing blade. It is found that the rub strip and mass eccentricity have the strongest influence on the LP shaft torsional loading. It is found that mass eccentricity has the largest influence on peak fan whirl. It is also found that released blade mass and attachment stiffness have the largest influence on the trailing blade loading.

scholarly journals Certified POD-Based Multiobjective Optimal Control of Time-Variant Heat Phenomena

2018 ◽  
Author(s):  
Stefan Banholzer ◽  
Eugen Makarov ◽  
Stefan Volkwein
Keyword(s):  
Optimal Control ◽  
Reference Point ◽  
Model Order Reduction ◽  
Optimization Problems ◽  
Order Reduction ◽  
Point Method ◽  
Model Order ◽  
Reference Point Method ◽  
Multiobjective Optimal Control ◽  
Proper Orthogonal

Many optimization problems in applications can be formulated using several objective functions, which are conflicting with each other. This leads to the notion of multiobjective or multicriterial optimization problems. Here, we investigate the application of the Euclidean reference point method in combination with model-order reduction to multiobjective optimal control problems. Since the reference point method transforms the multiobjective optimal control problem into a series of scalar optimization problems, the method of proper orthogonal decomposition (POD) is introduced as an approach for model-order reduction.

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