Part 1: Dynamical Characterization of a Frictionally Excited Beam

The dynamics of an experimental frictionally excited beam areinvestigated. The friction is characterized and shown to involve contactcompliance. Beam displacements are approximated from strain gagesignals. The system dynamics are rich, including a variety of periodic,quasi-periodic and chaotic responses. Proper orthogonal decomposition isapplied to chaotic data to obtain information about the spatialcoherence of the beam dynamics. Responses for different parameter valuesresult in a different set of proper orthogonal modes. The number ofproper orthogonal modes that account for 99.99% of the signalpower is compared to the corresponding number of linear normal modes,and it is verified that the proper orthogonal modes are more efficientin capturing the dynamics.     

The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview

Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New developments have been proposed in order to examine nonlinear systems, among which the theory based on nonlinear normal modes is indubitably the most appealing. In this paper, a different approach is adopted, and proper orthogonal decomposition is considered. The modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data. The utility of the method for dynamic characterization and order reduction of linear and nonlinear mechanical systems is demonstrated in this study.     

Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial

The proper orthogonal decomposition identifies basis functions or modes which optimally capture the average energy content from numerical or experimental data. By projecting the Navier–Stokes equations onto these modes and truncating, one can obtain low-dimensional ordinary differential equation models for fluid flows. In this paper we present a tutorial on the construction of such models. In addition to providing a general overview of the procedure, we describe two different ways to numerically calculate the modes, show how symmetry considerations can be exploited to simplify and understand them, comment on how parameter variations are captured naturally in such models, and describe a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently. We illustrate for the example of plane Couette flow in a minimal flow unit – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence.     

Characterization of the organization in shear layers via the Proper Orthogonal Decomposition

Experiments are performed in an incompressible plane turbulent mixing layer, using various hot wire rake configurations. From these experiments, the Proper Orthogonal Decomposition is applied for kernels where the space-time correlation tensor is evaluated over different spatial meshes and velocity components configurations. The resulting decompositions are then discussed in terms of characterization of the organization of the flow for various scalar or vectorial approaches of the POD. An incrtial range law is evidenced. The instantaneous contribution of the first modes of the POD to the organization of the flow is analyzed. A dynamical behavior for the organization of the flow is observed from the correlation between the first two modes contribution.     

Adaptive Feedback Linearizing Control of Proper Orthogonal Decomposition Nonlinear Flow Models

This paper treats the question of feedback linearizing control oftwo-dimensional incompressible, unsteady wake flow. For definiteness,flow past a circular cylinder is considered, but the design approachpresented here is applicable to other flow control problems. Twofinite-dimensional lower-order models based on Proper OrthogonalDecomposition (POD) of dimension N with N actuators are considered.Models I and II are obtained using control function and penalty functionmethods, respectively. Control action can be achieved by a combinationof suction, injection, and synthetic jets. For the design ofcontrollers, it is assumed that the system matrices of the POD modelsare unknown. Nonlinear adaptive control systems for the two models arederived. For model I, nontrivial zero-error dynamics exists, which playa key role in the stability of the closed-loop system. But for model II,global adaptive trajectory control is achieved. In the closed-loopsystem, the mode amplitudes asymptotically follow the referencetrajectories. Simulation results for a 4-mode POD model obtained usingthe penalty function method are presented. These results show that inthe closed-loop system, unsteadiness in the mode amplitudes can besuppressed in spite of large uncertainties in the flow model.     

Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods

Reduced order models for the dynamics of geometrically exact planar rods are derived by projecting the nonlinear equations of motion onto a subspace spanned by a set of proper orthogonal modes. These optimal modes are identified by a proper orthogonal decomposition processing of high-resolution finite element dynamics. A three-degree-of-freedom reduced system is derived to study distinct categories of motions dominated by a single POD mode. The modal analysis of the reduced system characterizes in a unique fashion for these motions, since its linear natural frequencies are near to the natural frequencies of the full-order system. For free motions characterized by a single POD mode, the eigen-vector matrix of the derived reduced system coincides with the principal POD-directions. This property reflects the existence of a normal mode of vibration, which appears to be close to a slow invariant manifold. Its shape is captured by that of the dominant POD mode. The modal analysis of the POD-based reduced order system provides a potentially valuable tool to characterize the spatio-temporal complexity of the dynamics in order to elucidate connections between proper orthogonal modes and nonlinear normal modes of vibration.     

Using a Proper Orthogonal Decomposition representation of the aerodynamic forces for stochastic buffeting prediction

High-performance aircraft often suffer from the consequences of tail buffeting at moderate subsonic Mach numbers and medium to high angles of attack. The impact of the aircraft’s highly unsteady flow field on the tails can result in significant structural fatigue and degraded handling qualities. Various methods have been developed to predict tail buffeting. Stochastic response methods are among frequently used approaches. For such methods the size of the excitation data set can become an issue, especially when the auto- and cross-spectra of all available excitation signals on the configuration are considered. The present paper demonstrates how to modify stochastic tail buffeting prediction methods using Proper Orthogonal Decomposition (POD). The approach is based on the modal decomposition of the aerodynamic buffet excitation data set. It notably reduces the computational effort for structural response and loads prediction with limited losses in accuracy while using all power- and cross-spectra of the reduced dataset. The method was applied to the computational buffeting prediction for a generic configuration with double-delta wing and horizontal tail plane (HTP) over a wide range of angles of attack. It was shown that the POD-modes of the aerodynamic buffet excitation resembled the characteristics of configuration’s complex vortical flow field. The predicted structural response and loads converged well with increasing number of POD-modes. With the presented approach, the computational effort of stochastic tail buffeting prediction has been reduced by orders of magnitude compared to the case with the full aerodynamic buffet excitation data set.     

广义边界梁的动力学建模与动态特征

对于广义边界条件Euler-Bernoulli梁,采用相对描述方式建立了可描述梁整体运动和相对变形的几何非线性及其线性化动力学模型,应用线性变换得到了该类梁的线性经典动力学方程,得到了广义边界条件下梁的横向振动代数特征方程、特征函数及特征值的退化表达式.算例分析了边界小扰动对固支-固支梁横向振动特征的影响规律.     

基于正交设计的冲击荷载下型钢混凝土梁动力性能有限元分析

为研究冲击荷载下型钢混凝土梁的动力性能,取试验结果验证模型有效性后,使用有限元软件ABAQUS建立了型钢混凝土梁落锤冲击模型．在对比了型钢混凝土梁与钢筋混凝土梁的动力性能后,分析了落锤速度对型钢混凝土梁的冲击位移、冲击力、冲击力位移曲线、惯性效应、塑性耗能等性能的影响,然后基于正交设计,研究了落锤质量、落锤速度、混凝土强度等参数对型钢混凝土梁动力性能的影响．研究结果表明：型钢混凝土梁的抗冲击性能较好;落锤速度是影响构件动力性能的主要参数,落锤质量对动力性能的影响较大,提高混凝土强度有助于增强构件的抗冲击性能．     

On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle

Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bernoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulomb's law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulomb's law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.The work of the last two authors was supported in part by Oakland University Research Fellowships.     

Dimension Reduction of Dynamical Systems: Methods, Models, Applications

After presenting some basic introductory ideas concerning dimension reduction and reduced order modelling, an overview of the contents of the papers collected in this Special Issue of Nonlinear Dynamics is given.     

The wetting problem of fluids on solid surfaces. Part 1: the dynamics of contact lines

The understanding of the spreading of liquids on solid surfaces is an important challenge for contemporary physics. Today, the motion of the contact line formed at the intersection of two immiscible fluids and a solid is still subject to dispute. In this paper, a new picture of the dynamics of wetting is offered through an example of non-Newtonian slow liquid movements. The kinematics of liquids at the contact line and equations of motion are revisited. Adherence conditions are required except at the contact line. Consequently, for each fluid, the velocity field is multivalued at the contact line and generates an equivalent concept of line friction but stresses and viscous dissipation remain bounded. A Young-Dupré equation for the apparent dynamic contact angle between the interface and solid surface depending on the movements of the fluid near the contact line is proposed.Received: 5 June 2001, Accepted: 24 May 2003, Published online: 29 July 2003PACS: 47.17., 47.50, 66.20, 68.08, 83.50     

考虑关节摩擦的空间机器人动力学建模与参数辨识

研究了考虑关节摩擦影响的空间机器人系统的动力学建模与参数辨识问题．采用单向递推组集方法和虚功率原理建立了含有关节摩擦的多体系统动力学方程,推导了关节摩擦对系统动力学方程的贡献,采用基于腕力传感器信号和最小二乘法的辨识方法进行了系统惯性参数的辨识．数值仿真结果验证了数学模型的正确性与辨识方法的有效性．     

The wetting problem of fluids on solid surfaces. Part 2: the contact angle hysteresis

In part 1 (Gouin, [13]), we proposed a model of dynamics of wetting for slow movements near a contact line formed at the interface of two immiscible fluids and a solid when viscous dissipation remains bounded. The contact line is not a material line and a Young-Dupré equation for the apparent dynamic contact angle taking into account the line celerity was proposed. In this paper we consider a form of the interfacial energy of a solid surface in which many small oscillations are superposed on a slowly varying function. For a capillary tube, a scaling analysis of the microscopic law associated with the Young-Dupré dynamic equation yields a macroscopic equation for the motion of the contact line. The value of the deduced apparent dynamic contact angle yields for the average response of the line motion a phenomenon akin to the stick-slip motion of the contact line on the solid wall. The contact angle hysteresis phenomenon and the modelling of experimentally well-known results expressing the dependence of the apparent dynamic contact angle on the celerity of the line are obtained. Furthermore, a qualitative explanation of the maximum speed of wetting (and dewetting) can be given.Received: 5 June 2001, Accepted: 24 May 2003, Published online: 29 July 2003PACS: 02.90, 47.50, 66.20, 68.03, 68.08     

Sampling and resolution characteristics in reduced order models of shallow water equations: Intrusive vs nonintrusive

We investigate the sensitivity of reduced order models (ROMs) to training data spatial resolution as well as sampling rate. In particular, we consider proper orthogonal decomposition (POD), coupled with Galerkin projection (POD-GP), as an intrusive ROM technique. For nonintrusive ROMs, we consider two frameworks. The first is using dynamic mode decomposition (DMD), and the second is based on artificial neural networks (ANNs). For ANN, we utilized a residual deep neural network, and for DMD we have studied two versions of DMD approaches; one with hard thresholding and the other with sorted bases selection. Also, we highlight the differences between mean subtracting the data (centering) and using the data without mean subtraction. We tested these ROMs using a system of 2D shallow water equations for four different numerical experiments, adopting combinations of sampling rates and spatial resolutions. For these cases, we found that the DMD basis obtained with hard thresholding is sensitive to sampling rate. The sorted DMD algorithm helps to mitigate this problem and yields more stabilized and converging solution. Furthermore, we demonstrate that both DMD approaches without mean subtraction provide significantly more accurate results than DMD with mean subtracting the data. On the other hand, POD is relatively insensitive to sampling rate and yields better representation of the flow field. Meanwhile, spatial resolution has little effect on both POD and DMD performances. Numerical results reveal that an ANN on POD subspace (POD-ANN) performs remarkably better than POD-GP and DMD in capturing system dynamics, even with a small number of modes.     

Wake structures and vortex-induced vibrations of a long flexible cylinder—Part 2: Drag coefficients and vortex modes

Drag coefficients and vortex structures in the wake of a vertical long flexible cylinder vibrating at low mode numbers are presented in this paper. A model with an external diameter of 16 mm and a total length of 1.5 m giving an aspect ratio of about 94 was used to perform more than 100 runs in which Reynolds numbers ranged between 1200 and 12 000. Only the lower 40% of its length was exposed to the water current in the flume and applied top tensions varied from 15 to 110 N giving fundamental natural frequencies in the range from 3 to 7.1 Hz. Reduced velocities based on the fundamental natural frequency up to 16 were reached. The mass ratio was 1.8 and the combined mass-damping parameter about 0.05. The largest drag coefficients were found related to the largest x–y synchronised motions. Digital particle image velocimetry was used to investigate the vortex structures in the wake of the cylinder model. Two modes of vortex shedding were observed, depending on the response branch and the position along the length of the model at which the interrogations were performed.     

Quadratic stochastic estimation of far‐field acoustic pressure with coherent structure events in a 2D compressible plane mixing layer

Mathematical tools based on cross correlations between aerodynamic quantities of interest inside the shear flow region and the radiated sound pressure are used to investigate noise generation mechanisms in a plane compressible mixing layer. An original methodology relying on an efficient coupling between proper orthogonal decomposition (POD) and stochastic estimation procedures is developed to analyze the main aerodynamic mechanisms that govern noise production. POD is used to split the instantaneous flow fluctuations as the sum of three components: the large‐ and small‐scale coherent structures (LCS and SCS) and the background quasi‐Gaussian fluctuations. Based on this flow partitioning, quadratic stochastic estimation is implemented to estimate the far‐field acoustic pressure associated with each flow component. The far field acoustic pressure associated with both LCS and SCS is then investigated. By analyzing the RMS and temporal spectra of the far‐field acoustic pressure, it is observed that the SCSs, as defined thanks to the POD basis, are responsible for the main part of the noise emission. Copyright © 2009 John Wiley & Sons, Ltd.     

Mathematical Modeling and Parameter Identification of a Planar Servo-Pneumatic Test Facility. Part I: Mathematical Modeling and Computer Simulation

High quality multi-axis test facilities used for testing heavy loads and large structures of industrial equipment are usually simulated, designed and controlled based on reduced model equations neglecting the inertia properties of the actuators. The design and control of servo-pneumatic test facilities used for testing small and light structures must take into account extended test facility models including the various inertia properties of the actuators. In this paper (Part I) an extended test facility model is presented including the various inertia properties and joints of the actuators. These extended model equations are represented in a form well suited to be directly implemented in control algorithms based on exact linearization techniques for real time control. This is done by stepwise projecting the inertia properties of the various actuator housings and actuator pistons down to the common mass of the test table and payload. The resulting extended model equations have the same form as the reduced model equations. They only include more complex system matrices and vector functions. These compact model equations turn out to be suitable for an efficient nonlinear controller design of these test facilities. Computer simulations and associated laboratory experiments show the necessity to use extended model equations in case of testing small and light structures. In Part II of this paper [1] the inertia parameters of the planar test facility will be identified in laboratory experiments.     

Near-critical CO2 liquid–vapor flow in a sub-microchannel. Part I: Mean-field free-energy D2Q9 lattice Boltzmann method

The mean-field free-energy based lattice Boltzmann method (LBM) is developed for the calculation of liquid–vapor flows in channels. We show that the extensively used common bounceback boundary condition leads to an unphysical velocity at the wall in the presence of surface forces that arise from any local forces such as gravity, fluid–fluid and fluid–solid interactions. We then develop a mass-conserving velocity-boundary condition which eliminates the unphysical velocities. An important aspect of the overall LBM model is the inclusion of the correct physics to simulate different wall wettabilities and dynamic contact lines. The model is applied to static and dynamic liquid–vapor interfacial flows and compared to theory. The model shows good agreement with three well established theories of contact line dynamics.     

General Spatial Dynamic Problem for an Elliptic Crack under the Action of a Normal Shear Wave,with Consideration for the Contact Interaction of the Crack Faces

Consideration is given to the contact interaction of the faces of a stationary plane elliptical crack under the action of a harmonic shear wave normally incident on the crack surface. The dependence of the mode II and III stress intensity factors on the wave number is studied for different values of the friction coefficient