Bilinear modal representations for reduced-order modeling of localized piecewise-linear oscillators

A novel reduced-order modeling method for vibration problems of elastic structures with localized piecewise-linearity is proposed. The focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity, such as cracked structures and delaminated plates. The modeling framework is based on observations of the proper orthogonal modes computed from nonlinear forced responses and their approximation by a truncated set of linear normal modes with special boundary conditions. First, it is shown that a set of proper orthogonal modes can form a good basis for constructing a reduced-order model that can well capture the nonlinear normal modes. Next, it is shown that the subspace spanned by the set of dominant proper orthogonal modes can be well approximated by a slightly larger set of linear normal modes with special boundary conditions. These linear modes are referred to as bi-linear modes, and are selected by an elaborate methodology which utilizes certain similarities between the bi-linear modes and approximations for the dominant proper orthogonal modes. These approximations are obtained using interpolated proper orthogonal modes of smaller dimensional models. The proposed method is compared with traditional reduced-order modeling methods such as component mode synthesis, and its advantages are discussed. Forced response analyses of cracked structures and delaminated plates are provided for demonstrating the accuracy and efficiency of the proposed methodology.     

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

Proper orthogonal decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter ν = 10^?3, the two-dimensional flow past a cylinder at Reynolds number Re = 200, and the three-dimensional flow past a cylinder at Reynolds number Re = 1000.     

A new measure of efficiency for model reduction: Application to a vibroimpact system

The aim of this paper is to propose a new way to measure the efficiency of the proper orthogonal decomposition (POD) to construct a reduced-order model in Structural Dynamics. It investigates the efficiency of three reduced-order models for a vibroimpact problem: (1) POD_dir-basis, which is the basis constructed using the direct method of the proper orthogonal decomposition; (2) POD_snap-basis, which is the basis constructed using the snapshot method of the POD; and (3) LIN-basis, which is the basis composed by the normal modes of the associated linear (LIN) conservative system. The efficiency is measured in terms of (1) number of elements to represent the dynamics with a given precision and (2) computational cost to simulate the time response within a given precision.     

Modeling vibratory damage with reduced-order models and the generalized finite element method

This paper investigates a coupled computational analysis framework that uses reduced-order models and the generalized finite element method to model vibratory induced stress near local defects. The application area of interest is the life prediction of thin gauge structural components exhibiting nonlinear, path-dependent dynamic response. Full-order finite element models of these structural components can require prohibitively large amounts of processor time. Recent developments in nonlinear reduced-order models have demonstrated efficient computation of the dynamic response. These models are relatively insensitive to small imperfections. Conversely, the generalized finite element method provides the ability to model local defects without geometric dependency on the mesh. A more robust version of the method, with numerically built enrichment functions, provides a multiple-scale modeling capability through direct coupling of global and local finite element models. Replacing the component finite element model with a reduced-order model allows for efficient computation of dynamic response while providing the necessary information to drive local, solid analyses which can zoom in on regions containing stress risers or cracks. This paper describes the coupling of these approaches to enable fatigue and crack propagation predictions. Numerical/experimental examples are provided.     

Modal reduction of mathematical models of biological molecules

This paper reports a detailed study of modal reduction based on either linear normal mode (LNM) analysis or proper orthogonal decomposition (POD) for modeling a single α-d-glucopyranose monomer as well as a chain of monomers attached to a moving atomic force microscope (AFM) under harmonic excitations. Also a modal reduction method combining POD and component modal synthesis is developed. The accuracy and efficiency of these methods are reported. The focus of this study is to determine to what extent these methods can reduce the time and cost of molecular modeling and simultaneously provide the required accuracy. It has been demonstrated that a linear reduced order model is valid for small amplitude excitation and low frequency excitation. It is found that a nonlinear reduced order model based on POD modes provides a good approximation even for large excitation while the nonlinear reduced order model using linear eigenmodes as the basis vectors is less effective for modeling molecules with a strong nonlinearity. The reduced order model based on component modal synthesis using POD modes for each component also gives a good approximation. With the reduction in the dimension of the system using these methods the computational time and cost can be reduced significantly.     

风力机翼型气动计算的降阶模型研究

将基于POD的降阶模型应用于风力机翼型的气动研究。首先应用CFD数值模拟得到一系列快照结果;应用基于本征正交分解(POD)的方法得到流场的一组基模态,认为对于所研究的问题,任一流场可以由这些基模态通过线性表达得到;对控制方程进行Galerkin投影,得到降阶模型,将离散求解N-S方程的问题转化为一组只有十几个自由度的常微分方程,从而减少计算时间,提高计算效率。并对二维翼型的绕流的定常和非定常问题进行了分析,计算结果表明,降阶模型可以较好地捕捉流动的特征,与直接CFD模拟相比计算精度相当,但大幅有效地提高了计算效率。     

Proper orthogonal decomposition of the dynamics in bolted joints

Joints play an important role in the dissipation of vibration energy in built-up structures. The highly nonlinear nature of joints with micro-slip is the main hurdle in developing a reduced-order model which can simulate the dynamic behaviour of a joint for a wide range of excitation conditions and geometries. In this paper, the proper orthogonal decomposition is applied to joint dynamics in an attempt to arrive at a generic reduced-order model without a compromise on the physics of the system. Only the linear part of the system of equations is reduced. The nonlinear part is determined in the full space and then reduced before the numerical integration phase. The major reduction in computational time is achieved by the increase in the size of the stable time step and the reduced number of coordinates in the integration phase. The reduced-order model, which is derived for an isolated and harmonically excited joint, is successfully applied to separate joints with different geometries and excitation conditions, e.g. harmonic and impulsive. The model is also capable of simulating the dynamics of structures with joints. The results of the reduced-order model show good agreement with a full model both in terms of the state of the system and its hysteretic behaviour.     

Proper orthogonal decomposition and galerkin projection for a three-dimensional plasma dynamical system

A general method by which to investigate nonlinear dynamical systems close to a stability threshold is presented. This method combines a proper orthogonal decomposition and a subsequent Galerkin projection. This technique is applied to three-dimensional resistive ballooning plasma fluctuations in a tokamak. The corresponding dynamical system belongs to a large family of convective fluid systems including Rayleigh-Benard convection. A proper orthogonal decomposition of the fluctuating signal obtained by numerical simulation shows that the relevant modes are close to the linear (global) modes. The Galerkin projection provides a low-dimensional system that allows the study of shear flow generation, its subsequent fluctuation reduction, and the evolution to oscillating states.     

基于POD模态分解的液环泵瞬态气液两相流分析

针对液环泵内气液两相流动的复杂时空规律,采用本征正交分解(POD)方法对其瞬态气液两相流场进行特征分解,分析其相态场、速度场的空间基模态特征及模态系数的时域特征,建立非定常流场降阶模型,并对流场进行预测分析。结果表明POD方法可实现对液环泵内复杂流场的时空解耦分析,相态场及速度场的各阶模态系数在时域内的变化能够反映各阶模态场的能量、频率及相位变化规律,模态场能够反映脉动流场的空间尺度变化规律。POD降阶模型能够对样本空间内的流场进行精确预测,进口压力、相态场及速度场预测结果的最大相对误差分别约为0.2%、4%、8%,在样本空间外POD降阶模型具有一定外延预测精度,当预测目标逐渐远离样本空间时POD预测结果与CFD计算结果之间的误差逐渐增大。     

ON PROPER ORTHOGONAL CO-ORDINATES AS INDICATORS OF MODAL ACTIVITY

Proper orthogonal decomposition (POD) is applied for examining modal activity. The extraction of proper orthogonal modal co-ordinates (POCs) is outlined. The proper orthogonal values (POVs) are the mean squared values of the POCs. The number of POVs above the noise floor provides a bound on the number of significant modes based on POVs above the noise floor. The ideas are illustrated on a linear cantilevered beam experiment. The displacement ensembles are obtained by processing six strain measurements. Coherent proper orthogonal modes (POMs) and POVs above the noise floor together confirm that six active modes are detected in the system. The distribution of modal components in POCs is discussed. The characteristics of the POMs, POVs and POCs are then examined in the presence of added noise.     

A time-varying identification method for mixed response measurements

The proper orthogonal decomposition is a method that may be applied to linear and nonlinear structures for extracting important information from a measured structural response. This method is often applied for model reduction of linear and nonlinear systems and has been applied recently for time-varying system identification. Although methods have previously been developed to identify time-varying models for simple linear and nonlinear structures using the proper orthogonal decomposition of a measured structural response, the application of these methods has been limited to cases where the excitation is either an initial condition or an applied load but not a combination of the two. This paper presents a method for combining previously published proper orthogonal decomposition-based identification techniques for strictly free or strictly forced systems to identify predictive models for a system when only mixed response data are available, i.e. response data resulting from initial conditions and loads that are applied together. This method extends the applicability of the previous proper orthogonal decomposition-based identification techniques to operational data acquired outside of a controlled laboratory setting. The method is applied to response data generated by finite element models of simple linear time-invariant, time-varying, and nonlinear beams and the strengths and weaknesses of the method are discussed.     

Vibration analysis of drillstrings with self-excited stick-slip oscillations

Drillstring vibration is one of the major causes for a deteriorated drilling performance. Field experience revealed that it is crucial to understand the complex vibrational mechanisms experienced by a drilling system in order to better control its functional operation and improve its performance. Sick-slip oscillations due to contact between the drilling bit and formation is known to excite severe torsional and axial vibrations in the drillstring. A dynamic model of the drillstring including the drillpipes and drillcollars is formulated. The equation of motion of the rotating drillstring is derived using Lagrangian approach in conjunction with the finite element method. The model accounts for the torsional-bending inertia coupling and the axial-bending geometric nonlinear coupling. In addition, the model accounts for the gyroscopic effect, the effect of the gravitational force field, and the stick-slip interaction forces. Explicit expressions of the finite element coefficient matrices are derived using a consistent mass formulation. The generalized eigenvalue problem is solved to determine modal transformations, which are invoked to obtain the reduced-order modal form of the dynamic equations. The developed model is integrated into a computational scheme to calculate time-response of the drillstring system in the presence of stick-slip excitations.     

近壁湍流的降阶模型及应用

由条带和流向涡的循环再生构成的近壁自维持过程(self-sustaining process, SSP)是壁湍流产生和维持的重要机制. 文章通过对最小槽道的直接数值模拟(direct numerical simulation, DNS)获得近壁自维持过程的流场数据, 采用正规正交分解法(proper orthogonal decomposition, POD)对该数据进行分析, 获得了不同流向和展向尺度的特征模态, 通过将Navier-Stokes方程在这些模态上进行投影, 得到近壁自维持过程的降阶模型, 并采用DNS数据对降阶模型的预测能力进行了评价. 该模型被初步应用于大涡模拟近壁模型的构造.      

Assessment of reduced models for the detection of modal interaction through rotor stator contacts

Interactions through direct contact between blade-tips and outer casings in modern turbomachines require complex formulations and subsequent expensive computational efforts when the classical finite element method is considered. The construction of reduced-order models through component mode synthesis techniques usually improves the computational efficiency and may be used for fast parameter studies yielding a better knowledge of the phenomena of interest.In this highly nonlinear framework, the present study is dedicated to the investigation of the capabilities of fixed- and free-interface reduction strategies to handle accurately such problems through a realistic 2D model and complements former results involving a direct modal projection with respective strong kinematic restrictions.The equations of motion are solved using an explicit time integration scheme together with the Lagrange multiplier method where friction is accounted for. The presented work discusses the notions of both displacement and motion convergences and the possibility to conduct fast parameter studies with the use of relevant reduction bases. It also shows that kinematic restrictions artificially enhance the detection of modal interactions.     

Efficient modal control strategies for active control of vibrations

Some efficient strategies for the active control of vibrations of a beam structure using piezoelectric materials are described. The control algorithms have been implemented for a cantilever beam model developed using finite element formulation. The vibration response of the beam to an impulse excitation has been calculated numerically for the uncontrolled and the controlled cases. The essence of the method proposed is that a feedback force in different modes be applied according to the vibration amplitude in the respective modes i.e., modes having lesser vibration may receive lesser feedback. This weighting may be done on the basis of either displacement or energy present in different modes. This method is compared with existing methods of modal space control, namely the independent modal space control (IMSC), and modified independent modal space control (MIMSC). The method is in fact an extension of the modified independent space control with the addition that it proposes to use the sum of weighted multiple modal forces for control. The proposed method results in a simpler feedback, which is easy to implement on a controller. The procedure is illustrated for vibration control of a cantilever beam. The analytical results show that the maximum feedback control voltage required in the proposed method is further reduced as compared to existing methods of IMSC and MIMSC for similar vibration control. The limitations of the proposed method are discussed.     

Projection-free proper orthogonal decomposition method for a cantilever plate in supersonic flow

This paper explores the use of the proper orthogonal decomposition (POD) method for supersonic nonlinear flutter of a cantilever plate or wing. The aeroelastic equations are constructed using von Karman plate theory and first-order piston theory. The two-dimensional POD modes (POMs) in xy plane are determined from the chaotic results given by the traditional Rayleigh–Ritz (RR) approach. For a specific structure, the POMs need to be calculated once and then can be used for various parameters of interest. The derivatives of the POMs are calculated numerically to avoid the complex projection from the POMs to the Rayleigh–Ritz modes (RRMs). Numerical examples demonstrate that the POD method using 4 POMs can obtain accurate limit cycle oscillation (LCO) results with substantial computational cost savings, compared with 12 RRMs by the Rayleigh–Ritz method. The POD method is employed for the analysis of the chaotic oscillations. It is also demonstrated that the POD modes are robust over a range of flight parameters.     

Analysis of nonlinear steady state vibration of a multi-degree-of-freedom system using component mode synthesis method

In this paper, an analytical approach for nonlinear forced vibration of a multi-degree-of-freedom system is proposed using the component mode synthesis method. The whole system is divided into some components and a nonlinear modal equation of each component is derived using the free-interface vibration modes. The modal equations of all components and the conjunction conditions are solved simultaneously, and then the modal responses of components are derived. Finally, the dynamic responses of the whole system can be obtained. The degrees of freedom of modal equations can be reduced when the lower vibration modes are only adopted in each component. As a numerical example, a nine-degree-of-freedom system is considered, in which all spring have cubic type nonlinearity. As a result, it is shown that when there are no rigid modes in components, the compliance by the proposed method agrees very well with the exact one even if the lower vibration modes of components are only adopted. The other hand, in the case with rigid modes in components, the compliance has a little error compared with the exact result. It is recognized that the method proposed is very effective in the case without rigid modes in components for the actual application.     

Reliable reduced-order models for time-dependent linearized Euler equations

Development of optimal reduced-order models for linearized Euler equations is investigated. Recent methods based on proper orthogonal decomposition (POD), applicable for high-order systems, are presented and compared. Particular attention is paid to the link between the choice of the projection and the efficiency of the reduced model. A stabilizing projection is introduced to induce a stable reduced-order model at finite time even if the energy of the physical model is growing. The proposed method is particularly well adapted for time-dependent hyperbolic systems and intrinsically skew-symmetric models. This paper also provides a common methodology to reliably reduce very large nonsymmetric physical problems.     

Modal interaction activations and nonlinear dynamic response of shallow arch with both ends vertically elastically constrained for two-to-one internal resonance

This study investigates the two-to-one internal resonance of the shallow arch with both ends elastically constraining, and the primary resonance case is considered. The full-basis Galerkin method and the multi-scale method are applied to obtain the modulation equations. It is shown that the natural frequencies of the first two modes cross/avoid to each other when the stiffness of elastic supports at two ends is the same/different. Moreover, the nonlinear modal interactions between these two modes may not/may be activated. The force/frequency-response curves are employed to explore the nonlinear response of the elastically supported shallow arch. The saddle-node bifurcation points and Hopf bifurcation points are observed in these cases. Moreover, the dynamic solutions, i.e., the periodic solution, quasi-periodic solution and chaotic solution are discussed. The numerical simulations are used to illustrate the route to chaos via period-doubling bifurcation.     

基于模型选择的模式波前重构算法研究

基于Zernike模式的波前重构算法通常忽略实际波前像差构成的差异, 而用一定数量的低阶Zernike模式进行波前重构, 导致模式混淆或耦合等问题, 进而影响波前重构的精度. 根据信息论中的最小描述长度准则对重构模型的阶数进行了选择, 在此基础上应用非线性优化算法计算重构系数, 并最终实现波前重构; 对不同信噪比条件下振幅均匀分布入射光束的波前进行了重构. 结果表明: 该算法不但能够实现相对于现有算法相对较高的波前重构精度, 并且具有优良的噪声适应性, 体现了模型选择在模式法波前重构算法中的意义.