Normal modes for piecewise linear vibratory systems

A method to construct the normal modes for a class of piecewise linear vibratory systems is developed in this study. The approach utilizes the concepts of Poincaré maps and invariant manifolds from the theory of dynamical systems. In contrast to conventional methods for smooth systems, which expand normal modes in a series form around an equilibrium point of interest, the present method expands the normal modes in a series form of polar coordinates in a neighborhood of an invariant disk of the system. It is found that the normal modes, modal dynamics and frequency-amplitude dependence relationship are all of piecewise type. A two degree of freedom example is used to demonstrate the method.     

Multidimensional Damage Identification Based on Phase Space Warping: An Experimental Study

A multidimensional damage identification scheme developed in previous work is modified and investigated experimentally. An experimental apparatus consists of a driven two-well magneto-elastic oscillator, where a cantilever beam vibrates in a magnetic potential field perturbed by two electromagnets. These electromagnets are activated by a computer controlled power supply and their terminal voltages are considered a two-dimensional damage variable. The effect of total change in the supply voltage of the electromagnets is approximately 4% shift in the experimentally measured natural frequencies of small oscillations in each well of the potential. Experimental runs are started in a nominally chaotic regime. The battery voltages are altered on specific trajectories in the damage (voltage) phase space. Damage identification is accomplished based on the elastic vibration data collected using laser vibrometers and a accelerometer. The phase space warping based damage tracking feature vectors are estimated using a new phase space partitioning scheme. The damage identification is achieved by applying smooth orthogonal decomposition to the obtained statistics. The effect of the data record size on the quality of reconstructed damage trajectory is investigated in a series of experiments. It is also demonstrated that the new partitioning scheme considerably improves signal-to-noise ration of the identified damage states.     

Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

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.     

Dynamical behaviors of nonlinear viscoelastic thick plates with damage

Based on the deformation hypothesis of Timoshenko's plates and the Boltzmann's superposition principles for linear viscoelastic materials, the nonlinear equations governing the dynamical behavior of Timoshenko's viscoelastic thick plates with damage are presented. The Galerkin method is applied to simplify the set of equations. The numerical methods in nonlinear dynamics are used to solve the simplified systems. It could be seen that there are plenty of dynamical properties for dynamical systems formed by this kind of viscoelastic thick plate with damage under a transverse harmonic load. The influences of load, geometry and material parameters on the dynamical behavior of the nonlinear system are investigated in detail. At the same time, the effect of damage on the dynamical behavior of plate is also discussed.     

A Variational Approach to Dynamics of Flexible Multibody Systems

Abstract

This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body reference frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed     

The method of manufactured solutions applied to chaotic systems

In this paper, the method of manufactured solutions is applied to dynamical systems possessing chaotic behavior. This method is used in a non-traditional way to identify points with potential numerical errors and to improve computational efficiency. The numerical errors may be due to the selection of error tolerances in the integration of the ordinary differential equations, computer arithmetic precision, etc. Two classical chaotic models and two ship capsize models are examined. The current approach has similarities to entrainment methods in chaotic control theory, where entrainment refers to two dynamical systems having the same period, phase and amplitude. The convergent region from control theory is used to give a rough guide for identifying potentially catastrophic numerical errors for the classical chaotic models. The effectiveness of this method in improving computational efficiency is demonstrated for the ship capsize models.     

SPATIO-TEMPORAL CHAOTIC SYNCHRONIZATION FOR MODES COUPLED TWO GINZBURG-LANDAU EQUATIONS

On the basis of numerical computation, the conditions of the modes coupling are proposed, and the high-frequency modes are coupled, but the low frequency modes are uncoupled. It is proved that there exist an absorbing set and a global finite dimensional attractor which is compact and connected in the function space for the high-frequency modes coupled two Ginzburg-Landau equations(MGLE). The trajectory of driver equation may be spatio-temporal chaotic. One associates with MGLE, a truncated form of the equations. The prepared equations persist in long time dynamical behavior of MGLE. MGLE possess the squeezing properties under some conditions. It is proved that the complete spatio-temporal chaotic synchronization for MGLE can occur. Synchronization phenomenon of infinite dimensional dynamical system (IFDDS) is illustrated on the mathematical theory qualitatively. The method is different from Liapunov function methods and approximate linear methods.     

A new way to compute the Lyapunov characteristic exponents for non-smooth and discontinues dynamical systems

Nonlinear Dynamics - One of the most important problems of nonlinear dynamics is related to the development of methods concerning the identification of the dynamical modes of the corresponding...     

On the stability and extension of reduced-order Galerkin models in incompressible flows

Proper orthogonal decomposition (POD) has been used to develop a reduced-order model of the hydrodynamic forces acting on a circular cylinder. Direct numerical simulations of the incompressible Navier–Stokes equations have been performed using a parallel computational fluid dynamics (CFD) code to simulate the flow past a circular cylinder. Snapshots of the velocity and pressure fields are used to calculate the divergence-free velocity and pressure modes, respectively. We use the dominant of these velocity POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier–Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. The solution of the reduced dynamical system is a limit cycle corresponding to vortex shedding. We investigate the stability of the limit cycle by using long-time integration and propose to use a shooting technique to home on the system limit cycle. We obtain the pressure-Poisson equation by taking the divergence of the Navier–Stokes equation and then projecting it onto the pressure POD modes. The pressure is then decomposed into lift and drag components and compared with the CFD results.     

On the hidden beauty of the proper orthogonal decomposition

The proper orthogonal decomposition theorem (Loève, 1955) of probability theory has been proposed by Lumley (1967, 1972, 1981) for detection of spatial coherent patterns in turbulent flows. More specifically, the decomposition extracts deterministic functions from second-order statistics of a random field and converges optimally fast in quadratic mean (i.e., in energy). The technique can be made completely deterministic in the sense that it can be applied to spatially and temporally evolving flows. The remarkable property of this deterministic decomposition is not only in its optimal convergence (as emphasized before) but also in its space/time symmetry which permits access to the spatiotemporal dynamics. The flow is decomposed into both spatial and temporal orthogonal modes which are coupled: each space component is associated with a time component partner. The latter is the time evolution of the former and the former is the spatial configuration of the latter. This generalizes the notion of spatial and temporal structures which can be followed through the various instabilities that the flow undergoes as Reynolds number increases. It also provides a nonlinear dynamics tool for spatiotemporal dynamical systems and can be used for bifurcation detection and analysis as well as dimension and degree of complexity estimates.Dedicated to Professor J.L. Lumley on the occasion of his 60th birthday.This work was supported by an NSF/PYI award MSS89-57462, and partially by a NATO Grant No. 900265 which are gratefully acknowledged.     

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.     

Rich dynamics caused by diffusion

It is an awfully difficult task to design an efficient numerical method for bifurcation diagrams, the graphs of Lyapunov exponents, or the topological entropy about discrete dynamical systems by linear/nonlinear diffusion with the Direchlet/Neumann- boundary conditions. Until now there are less works concerned with such a problem. In this paper, we propose a scheme about bifurcating analysis in a series of discrete-time dynamical systems with linear/nonlinear diffusion terms under the periodic boundary conditions. The complexity of dynamical behaviors caused by the diffusion term are to be determined. Bifurcation diagrams are shown by numerical simulation and chaotic behavior (chaotic Turing patterns) is demonstrated by computing the largest Lyapunov exponent. Our theoretical model can give an interesting case study about the phenomenon: the individuals exhibit a very simple dynamics but the groups with linear/nonlinear coupling can own a complex dynamics including fluctuation, periodicity and even chaotic behavior. We find that diffusion can trigger chaotic behavior in the present system and there exist multiple Turing patterns. It is interesting as regular or chaotic patterns can be reported in this study. Chaotic orbits emerge when exploring further in the diffusion coefficient space, and such a behavior is entirely absent in the corresponding continuous time-space system. The method proposed in the present paper is innovative and the conclusion is novel.

     

An intelligent parameter varying (IPV) approach for non-linear system identification of base excited structures

Health monitoring and damage detection strategies for base-excited structures typically rely on accurate models of the system dynamics. Restoring forces in these structures can exhibit highly non-linear characteristics, thus accurate non-linear system identification is critical. Parametric system identification approaches are commonly used, but require a priori knowledge of restoring force characteristics. Non-parametric approaches do not require this a priori information, but they typically lack direct associations between the model and the system dynamics, providing limited utility for health monitoring and damage detection. In this paper a novel system identification approach, the intelligent parameter varying (IPV) method, is used to identify constitutive non-linearities in structures subject to seismic excitations. IPV overcomes the limitations of traditional parametric and non-parametric approaches, while preserving the unique benefits of each. It uses embedded radial basis function networks to estimate the constitutive characteristics of inelastic and hysteretic restoring forces in a multi-degree-of-freedom structure. Simulation results are compared to those of a traditional parametric approach, the prediction error method. These results demonstrate the effectiveness of IPV in identifying highly non-linear restoring forces, without a priori information, while preserving a direct association with the structural dynamics.     

Control of a separated boundary layer: reduced-order modeling using global modes revisited

The possibility of model reduction using global modes is readdressed, aiming at the controlling of a globally unstable separation bubble induced by a bump geometry. A combined oblique and orthogonal projection approach is proposed to design an estimator and controller in a Riccati-type feedback setting. An input?Coutput criterion is used to appropriately select the modes of the projection basis. The full-state linear instability dynamics is shown to be successfully controlled by the feedback coupling with controllers of moderate degrees of freedom.     

Multi-time-delay LSE-POD complementary approach applied to unsteady high-Reynolds-number near wake flow

This investigation compared the application and accuracy of single- and multi-time-delay linear stochastic estimation-proper orthogonal decomposition (LSE-POD) methods in the temporal domain. These methods were considered for low-dimensional estimations of the dynamics of the energy-containing structures in a high Reynolds number flow. The near wake dynamics of a bluff body were used to demonstrate the robustness and accuracy of the investigated LSE-POD methods. Statistically independent two-dimensional particle image velocimetry (PIV) measurements were used to determine spatial POD modes, and time-resolved surface pressure measurements were used to determine LSE coefficients required for estimating the time-varying POD coefficients. A low-order, time-resolved reconstruction of the wake dynamics was accomplished using these estimated time-varying POD coefficients. The paper also provides details concerning the accuracy of the estimation using multi-time-delay LSE-POD. The results demonstrate that the multi-time LSE-POD technique is successful in capturing and reconstructing the important near wake dynamics. It is also shown that optimizing the time delays used for the estimations increases the accuracy of the reconstruction. As a result of its capabilities, the multi-time-delay implementation of the LSE-POD approach offers an alternate method for low-dimensional modeling that is attractive for real-time flow estimation.     

Invariant Manifolds,Nonclassical Normal Modes,and Proper Orthogonal Modes in the Dynamics of the Flexible Spherical Pendulum

It is shown that the flexible spherical pendulum undergoes purely slow motions with master and slaved components. The family of slow motions is realized as a three-dimensional invariant manifold in phase space. This manifold is computed analytically by applying the method of geometric singular perturbations. This manifold is nonlinear and for all energy and angular momentum levels is characterized precisely by three PO (proper orthogonal) modes. Its submanifold of zero angular momentum is a two-dimensional invariant manifold; it is the geometric realization of a nonclassical nonlinear normal mode. This normal mode is characterized precisely by two PO modes. The slaved slow dynamics are characterized precisely by a single PO mode. The stability of the slow invariant manifold as well as interactions between fast and slow dynamics are considered.     

空间可展机构非光滑力学模型和动力学研究

空间可展机构广泛应用于展开和支撑柔性太阳能帆板和航天工程领域中的有效载荷, 包括抛物面天线、平面相控阵雷达和合成孔径雷达等. 非光滑特性及其相应的动力学现象在空间可展机构的设计中有着非常重要的作用. 该文系统地综述了空间可展机构非光滑力学建模与非线性动力学的研究进展. 首先详细描述了含间隙铰链的接触碰撞力和摩擦力等非光滑特点;然后系统地介绍了含间隙机构的动力学建模方法、分析方法和参数设计;进一步简单介绍了含间隙铰链空间可展机构的非线性动力学特性, 如谐波共振、周期运动的稳定性和各类分岔等;最后提出了空间可展机构非光滑动力系统动力学、稳定性与控制中亟待解决的若干问题.