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.     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: Comparison of POD and asymptotic nonlinear normal modes methods

The aim of the present paper is to compare two different methods available for reducing the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD), and an asymptotic approximation of the nonlinear normal modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the partial differential equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed. The response is investigated also for a fixed excitation frequency by using the excitation amplitude as bifurcation parameter for a wide range of variation. Bifurcation diagrams of Poincaré maps obtained from direct time integration and calculation of the maximum Lyapunov exponent have been used to characterize the system.     

Performance of Nonlinear Vibration Absorbers for Multi-Degrees-of-Freedom Systems Using Nonlinear Normal Modes

Linear vibration absorbers are a valuable tool used to suppressvibrations due to harmonic excitation in structural systems. Whilelimited evaluation of the performance of nonlinear vibrationabsorbers for nonlinear structures exists in the literature forsingle mode structures, none exists for multi-mode structures.Consequently, nonlinear multiple-degrees-of-freedom structures areevaluated. The theory of nonlinear normal modes is extended toinclude consideration of modal damping, excitation and smalllinear coupling, allowing estimation of vibration absorberperformance. The dynamics of the N +1-degrees-of-freedom system areshown to reduce to those of a two-degrees-of-freedom system on afour-dimensional nonlinear modal manifold, thereby simplifying theanalysis. Quantitative agreement is shown to require a higher-order model which is recommended for future investigation.     

Mode identification of a rotating disk

A modal testing method permitting identification of the natural frequencies, the number of nodal diameters and wave motions in a rotating disk is presented in this paper. This method is applicable at arbitrary rotation speed without requiring a priori information about the vibration modes of the stationary disk. The influence of disk rotation speed on the prediction of mode shapes with this method is shown, and experimental predictions of modal parameters are presented for both axisymmetric and asymmetric disks.     

Global bifurcations and chaotic motions of a flexible multi-beam structure

Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.     

Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder

This paper presents reduced order modelling (ROM) in fluid–structure interaction (FSI). The ROM via the proper orthogonal decomposition (POD) method has been chosen, due to its efficiency in the domain of fluid mechanics. POD-ROM is based on a low-order dynamical system obtained by projecting the nonlinear Navier–Stokes equations on a smaller number of POD modes. These POD modes are spatial and temporally independent. In FSI, the fluid and structure domains are moving, owing to which the POD method cannot be applied directly to reduce the equations of each domain. This article proposes to compute the POD modes for a global velocity field (fluid and solid), and then to construct a low-order dynamical system. The structure domain can be decomposed as a rigid domain, with a finite number of degrees of freedom. This low-order dynamical system is obtained by using a multiphase method similar to the fictitious domain method. This multiphase method extends the Navier–Stokes equations to the solid domain by using a penalisation method and a Lagrangian multiplier. By projecting these equations on the POD modes obtained for the global velocity field, a nonlinear low-order dynamical system is obtained and tested on a case of high Reynolds number.     

Aeroelastic effect on modal interaction and dynamic behavior of acoustically excited metallic panels

This paper details the study of the aeroelastic effect on modal interaction and dynamic behavior of acoustically excited square metallic panels with fully clamped edges using finite element method. The first-order shear deformation plate theory and von Karman nonlinear strain–displacement relationships are employed to consider the structural geometric nonlinearity caused by large vibration deflections. Piston aerodynamic theory and Gaussian white noise are used to simulate the aerodynamic load and the acoustic load, respectively. Motion equations are derived by the principle of virtual work in the physical coordinates and then transformed into the truncated modal coordinates with reduced orders. Runge–Kutta method is employed to obtain the system response, and the modal interaction mechanism is quantitatively valued by the modal participation distribution. Results show that in the pre-/near-flutter regions, in addition to the dominant fundamental resonant mode, the first twin companion antisymmetric modes can be largely excited by the aeroelastic coupling mechanism; thus, aeroelastic modal participation distribution and the spectrum response can be altered, while the dynamic behavior still exhibits linear random vibrations. In the post-flutter region, the dominant flutter motion can be enriched by highly ordered odd order super-harmonic motion occurs due to 1:1 internal resonances. Correspondingly, the panel dynamic behavior changes from random vibration to highly ordered motions in the fashion of diffused limit-cycle oscillations (LCOs). However, this LCOs motion can be affected by the intensifying acoustic excitation through changing the aeroelastic modal interaction mechanism. Accompanied with these changes, the panel can experience various stochastic bifurcations.     

Two modes nonresonant interaction for rectangular plate with geometrical nonlinearity

The vibrations of thin rectangular plate with geometrical nonlinearity are analyzed. The models of plate vibrations with different numbers of degrees-of-freedom are derived. It is deduced that two degrees-of-freedoms are enough to describe low-frequency nonlinear dynamics of plates. Nonlinear normal modes are used to analyze the system dynamics. If vibrations amplitudes are increased, single-mode plate vibrations are transformed into two mode ones. In this case, internal resonance conditions are not observed. Such transformation of vibration is described using Kauderer?CRosenberg nonlinear normal modes.     

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.     

DYNAMICAL CHARACTERISTICS ANALYSIS FOR UNILATERAL CONSTRAINED SIMPLY SUPPORTED BEAM WITH HARMONIC WAVE EXCITATION1)

为研究间隙碰撞对系统动力学响应的影响,以理想简支梁的振型函数为Rayleigh--Ritz基函数建立了单侧约束简支梁系统的非线性离散动力学方程组,应用数值方法研究了系统在基础谐波激励下的动力学响应特征及其对共振频率线性等效方法适用性的影响。研究表明：非线性动力学系统间隙产生的局部碰撞,使得系统振动能量在系统各阶模态之间转移,使得线性等效方法失效;即使进行非线性分析,也需要考虑系统固有频率远大于激励频段上限的模态。     

Part 2: Proper Orthogonal Modal Modeling of a Frictionally Excited Beam

Proper orthogonal modes (POMs) are used for order reduction ina beam with frictional excitation. The distributed model is based on anEuler–Bernoulli beam with frictional excitation. The friction ismodeled with Coulomb's law and contact compliance, and the contactsurface undergoes an imposed oscillation. The POMs are selected from achaotic response to build the reduced system model. These POMs areextrapolated into proper orthogonal modal functions (POMFs) by using thelinear normal modes as basis functions. The POMFs are used as the basisfor projection the partial differential equation of motion to alow-order set of ordinary differential equations. Simulated responsesbased on the POMFs and linear normal modes are compared to that of a'truth set' simulation, which is based on ten linear normal modes.     

Nonlinear coupled vibration of electrostatically actuated clamped–clamped microbeams under higher-order modes excitation

Nonlinear modal interactions have recently become the focus of intense research in micro-resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. Understanding and controlling nonlinear coupling between vibrational modes is critical for the development of advanced micromechanical devices. This article aims to theoretically investigate the influence of antisymmetry mode on nonlinear dynamic characteristics of electrically actuated microbeam via considering nonlinear modal interactions. Under higher-order modes excitation, two nonlinear coupled flexural modes to describe microbeam-based resonators are obtained by using Hamilton’s principle and Galerkin method. Then, the Method of Multiple Scales is applied to determine the response and stability of the system for small amplitude vibration. Through Hopf bifurcation analysis, the bifurcation sets for antisymmetry mode vibration are theoretically derived, and the mechanism of energy transfer between antisymmetry mode and symmetry mode is detailed studied. The pseudo-trajectory processing method is introduced to investigate the influence of external drive on amplitude and bifurcation behavior. Results show that nonlinear modal interactions can transit vibration energy from one mode to nearby mode. In what follows, an effective way is proposed to suppress midpoint displacement of the microbeam and to reduce the possibility of large deflection. The quantitative relationship between vibrational modes is also obtained. The displacement of one mode can be predicted by detecting another mode, which shows great potential of developing parameter design in MEMS. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.     

Nonlinear vibrations of a circular cylindrical shell with multiple internal resonances under multi-harmonic excitation

The nonlinear response of a water-filled, thin circular cylindrical shell, simply supported at the edges, to multi-harmonic excitation is studied. The shell has opportune dimensions so that the natural frequencies of the two modes (driven and companion) with three circumferential waves are practically double than the natural frequencies of the two modes (driven and companion) with two circumferential waves. This introduces a one-to-one-to-two-to-two internal resonance in the presence of harmonic excitation in the spectral neighbourhood of the natural frequency of the mode with two circumferential waves. Since the system is excited by a multi-harmonic point-load excitation composed by first and second harmonics, very complex nonlinear dynamics is obtained around the resonance of the fundamental mode. In fact, at this frequency, both modes with two and three circumferential waves are driven to resonance and each one is in a one-to-one internal resonance with its companion mode. The nonlinear dynamics is explored by using bifurcation diagrams of Poincaré maps and time responses.     

Ritz-POD法的原理及应用

提出一种在频域内分析大跨度屋盖结构的风振响应的Ritz-POD法。该方法用本征正交分解法(POD法)分解脉动风压场,得到占优的前几阶本征模态作为荷载的空间分布形式;根据此荷载空间分布形式生成Ritz向量,用Ritz向量直接叠加法分析结构风振响应,以解决大跨度屋盖结构多振型参与振动且高阶振型对结构响应贡献可能较大这一问题。运用该方法分析一单层球面网壳的风振响应,并与振型分解法对比,结果表明Ritz-POD法仅用少量的Ritz向量进行结构风振响应分析便可达到较高精度。     

温度变化对端部激励斜拉索共振响应影响

基于增量热场理论,利用Hamilton变分原理,通过引入与张拉力和垂度相关的无量纲参数,建立了考虑温度变化影响下斜拉索非线性动力学模型,并推导其面内/外非线性运动微分方程。考虑斜拉索受端部激励,利用Galerkin法得到离散后的无穷维常微分方程组。面内和面外运动各取前两阶模态,向前和向后扫频,利用龙格-库塔法数值积分求解常微分方程组,得到共振区域的幅频响应曲线。算例分析表明,温度变化和斜拉索固有频率呈反比例关系;温度变化会导致斜拉索共振特性发生定性和定量的改变,如共振区间发生漂移、跳跃点位置发生移动、共振响应幅值发生改变;端部位移激励下,温度变化有可能导致斜拉索更多模态受到激发,从而影响各阶模态的能量以及模态间的能量传递。     

On the use of sensitivity analysis in model reduction to predict flows for varying inflow conditions

The proper orthogonal decomposition (POD)‐based model reduction method is more and more successfully used in fluid flows. However, the main drawback of this methodology rests in the robustness of these reduced order models (ROMs) beyond the reference at which POD modes have been derived. Any variation in the flow or shape parameters within the ROM fails to predict the correct dynamics of the flow field. To broaden the spectrum of these models, the POD modes should have the global characteristics of the flow field over which the predictions are required. Mixing of snapshots with varying parameters is one way to improve the global nature of the modes but is computationally demanding because it requires full‐order solutions for a number of parameter values in order to assemble atextitrich enough database on which to perform POD. Instead, we have used sensitivity analysis (SA) to include the flow and shape parameters influence during the basis selection process to develop more robust ROMs for varying viscosity (Reynolds number), changing orientation and shape definition of bodies. This study aims at extending these ideas to inflow conditions to demonstrate the effectiveness of the proposed approach in capturing the effect of varying inflow on the dynamics of the flow over an elliptic cylinder. Numerical experiments show that the newly derived models allow for a more accurate representation of the flows when exploring the parameter space. Copyright © 2011 John Wiley & Sons, Ltd.     

Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint

Analytical results are presented on chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. The L-shaped beam is composed of two beams which are a horizontal beam and a vertical beam. The two beams are firmly connected with a right angle at each end. The beams joint with the right angle is attached to a linear spring. The other ends are firmly clamped for displacement. The L-shaped beam is compressed horizontally via the spring at the beams joint. The L-shaped beam deforms to a post-buckled configuration. Boundary conditions are required with geometrical continuity of displacements and dynamical equilibrium with axial force, bending moment, and share force, respectively. In the analysis, the mode shape function proposed by the senior author is introduced. The coefficients of the mode shape function are fixed to satisfy boundary conditions of displacements and linearized equilibrium conditions of force and moment. Assuming responses of the beam with the sum of the mode shape function, then applying the modified Galerkin procedure to the governing equations, a set of nonlinear ordinary differential equations is obtained in a multiple-degree-of-freedom system. Nonlinear responses of the beam are calculated under periodic lateral acceleration. Nonlinear frequency response curves are computed with the harmonic balance method in a wide range of excitation frequency. Chaotic vibrations are obtained with the numerical integration in a specific frequency region. The chaotic responses are investigated with the Fourier spectra, the Poincaré projections, the maximum Lyapunov exponents and the Lyapunov dimension. Applying the procedure of the proper orthogonal decomposition to the chaotic responses, contribution of vibration modes to the chaotic responses is confirmed. The following results have been found: The chaotic responses are generated with the ultra-subharmonic resonant response of the two-third order corresponding to the lowest mode of vibration. The Lyapunov dimension shows that three modes of vibration contribute to the chaotic vibrations predominantly. The results of proper orthogonal decomposition confirm that the three modes contribute to the chaos, which are the first, second, and third modes of vibration. Moreover, the results of the proper orthogonal decomposition are evaluated with velocity which is equivalent to kinetic energy. Higher modes of vibration show larger contribution to the chaotic responses, even though the first mode of vibration has the largest contribution ratio.     

利用多维模态理论分析圆柱贮箱液体非线性晃动

将多维模态理论应用到求解作横向运动圆柱贮箱中液体的非线性晃动问题. 首先通过压力积分变分原理推导出描述液体作非线性晃动的一般形式无穷维模态系统,然后根据Narimanov-Moiseev三阶渐近假设关系,通过选取二阶主模态和三阶次模态,将无穷维模态系统降为五维渐近模态系统. 通过对这个模态系统的数值积分可以看出一些典型的非线性特征(如波峰大于波谷、节径移动等).