Order reduction and nonlinear behaviors of a continuous rotor system

Ding, Qian; Zhang, Xiaoying

doi:10.1007/s11071-011-9975-8

Cited by 11 publications

(4 citation statements)

References 24 publications

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“…The approximate inertial manifold (AIM) provides an approximation of the solution of a PDE over sufficiently long time scales, using the theory of inertial manifolds. This is done by enslaving the amplitudes of fast modes in the system as a smooth graph over the amplitudes of a few selected modes while assuming that the corresponding inertial forces are not excited (see [26,6], for instance). The so called static condensation approach (cf.…”

Section: Nonlinear Manifolds In Model Reductionmentioning

confidence: 99%

Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

Jain

Tiso

2019

Journal of Computational and Nonlinear Dynamics

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Common trends in model reduction of large nonlinear finite-element-discretized systems involve Galerkin projection of the governing equations onto a low-dimensional linear subspace. Though this reduces the number of unknowns in the system, the computational cost for obtaining the reduced solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then used for fast approximation of these nonlinear terms. In the finite-element context, the energy conserving sampling and weighing (ECSW) method has emerged as an effective tool for hyper-reduction of Galerkinprojection-based reduced-order models (ROM).More recent trends in model reduction involve the use of nonlinear manifolds, which involves projection onto the tangent space of the manifold. While there are many methods to identify such nonlinear manifolds, hyper-reduction techniques to accelerate computation in such ROMs are rare. In this work, we propose an extension to ECSW to allow for hyper-reduction using nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics have been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.

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Section: Nonlinear Manifolds In Model Reductionmentioning

confidence: 99%

Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

Jain

Tiso

2019

Journal of Computational and Nonlinear Dynamics

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“…For example, Wang and Cao [15,16] proposed the predictive correction Galerkin method, and they used the method to reduce the order of the large rotor-bearing system. Ding and Zhang [52] analyzed an isotropic flexible shaft acted by nonlinear fluid-induced forces and reduced dimensions of the rotor system by both the standard Galerkin method and the nonlinear Galerkin method. An adaptive discontinuous Galerkin method was proposed to obtain a high-resolution numerical solution efficiently in Ref.…”

Section: Galerkin Methodsmentioning

confidence: 99%

A Review of Model Order Reduction Methods for Large‐Scale Structure Systems

Zhang

et al. 2021

Shock and Vibration

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The large-scale structure systems in engineering are complex, high dimensional, and variety of physical mechanism couplings; it will be difficult to analyze the dynamic behaviors of complex systems quickly and optimize system parameters. Model order reduction (MOR) is an efficient way to address those problems and widely applied in the engineering areas. This paper focuses on the model order reduction of high-dimensional complex systems and reviews basic theories, well-posedness, and limitations of common methods of the model order reduction using the following methods: center manifold, Lyapunov–Schmidt (L-S), Galerkin, modal synthesis, and proper orthogonal decomposition (POD) methods. The POD is a powerful and effective model order reduction method, which aims at obtaining the most important components of a high-dimensional complex system by using a few proper orthogonal modes, and it is widely studied and applied by a large number of researchers in the past few decades. In this paper, the POD method is introduced in detail and the main characteristics and the existing problems of this method are also discussed. POD is classified into two categories in terms of the sampling and the parameter robustness, and the research progresses in the recent years are presented to the domestic researchers for the study and application. Finally, the outlooks of model order reduction of high-dimensional complex systems are provided for future work.

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“…Currently, although many discrete methods are used to transform the diffusion-reaction PDE of eq into the ODE, − they require a known PDE and pay less attention to the time-varying dynamics of the DPS, which causes them to be less effective for modeling of an unknown and time-varying DPS. Most existing data-driven modeling methods are linear; thus, nonlinear spatial variations are ignored.…”

Section: Problem Descriptionmentioning

confidence: 99%

“…This research may be classified into two categories: one where the partial differential equation (PDE) of the DPS is known and one where the PDE is unknown. For the former, many approaches were developed to transform the PDE into an ordinary differential equation (ODE) in order to easily predict the dynamics of the DPS and design the controller of the DPS. − The finite difference method (FDM), , the finite element method (FEM), , and the method of multiple scales , are the most common methods used in this transformation. These methods discretize the PDE into the ODE.…”

Section: Introductionmentioning

confidence: 99%

Online Spatiotemporal Least-Squares Support Vector Machine Modeling Approach for Time-Varying Distributed Parameter Processes

Yin

Huang

2017

Ind. Eng. Chem. Res.

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Nonlinear and time-varying distributed parameter systems (DPSs) are challenging to accurately model due to potential spatiotemporal coupling, infinite-dimensional property, and time-varying dynamics. Although the least-squares support vector machine (LS-SVM) can effectively model lumped parameter systems, it is less effective to model the time-varying dynamics of a DPS, as it lacks the ability to incorporate time-varying spatiotemporal dynamics. Here, an online spatiotemporal LS-SVM approach is proposed to model a nonlinear and time-varying DPS. An adaptive spatial kernel function is first developed for online capture of the time-varying relationship between spatial locations. An online time coefficient model is then constructed to account for the time-varying temporal dynamics of the DPS. Combination of the adaptive spatial kernel function with the online time coefficient model allows for reconstruction of the complex DPS and ensures the model can reflect real-time spatiotemporal dynamics well. Experiments on a laboratory curing thermal process show the effectiveness of the proposed method.

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Order reduction and nonlinear behaviors of a continuous rotor system

Cited by 11 publications

References 24 publications

Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

A Review of Model Order Reduction Methods for Large‐Scale Structure Systems

Online Spatiotemporal Least-Squares Support Vector Machine Modeling Approach for Time-Varying Distributed Parameter Processes

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