Fluid Flow-Induced Nonlinear Vibration of Suspended Cables

The nonlinear interaction of the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable is studied. The governing equations of motion for two-mode interaction are derived on the basis of a general continuum model. The interaction causes the modal differential equations of the cable to be non-self-adjoint. As the flow speed increases above a certain critical value, the cable experiences oscillatory motion similar to the flutter of aeroelastic structures. A co-ordinate transformation in terms of the transverse and stretching motions of the cable is introduced to reduce the two nonlinearly coupled differential equations into a linear ordinary differential equation governing the stretching motion, and a strongly nonlinear differential equation for the transverse motion. For small values of the gravity-to-stiffness ratio the dynamics of the cable is examined using a two-time-scale approach. Numerical integration of the modal equations shows that the cable experiences stretching oscillations only when the flow speed exceeds a certain level. Above this level both stretching and transverse motions take place. The influences of system parameters such as gravity-to-stiffness ratio and density ratio on the response characteristics are also reported.     

Towards linear modal analysis for an L-shaped beam: Equations of motion

We consider an L-shaped beam structure and derive all the equations of motion considering also the rotary inertia terms. We show that the equations are decoupled in two motions, namely the in-plane bending and out-of-plane bending with torsion. In neglecting the rotary inertia terms the torsional equation for the secondary beam is fully decoupled from the other equations for out-of-plane motion. A numerical modal analysis was undertaken for two models of the L-shaped beam, considering two different orientations of the secondary beam, and it was shown that the mode shapes can be grouped into these two motions: in-plane bending and out-of-plane motion. We compared the theoretical natural frequencies of the secondary beam in torsion with finite element results which showed some disagreement, and also it was shown that the torsional mode shapes of the secondary beam are coupled with the other out-of-plane motions. These findings confirm that it is necessary to take rotary inertia terms into account for out-of-plane bending. This work is essential in order to perform accurate linear modal analysis on the L-shaped beam structure.     

High-dimensional time series prediction using kernel-based Koopman mode regression

This paper investigates the nonlinear dynamics of a doubly clamped piezoelectric nanobeam subjected to a combined AC and DC loadings in the presence of three-to-one internal resonance. Surface effects are taken into account in the governing equation of motion to incorporate the associated size effects at nanoscales. The reduced-order model equation (ROM) is obtained based on the Galerkin method. The multiple scales method is applied directly to the nonlinear equation of motion and associated boundary conditions to obtain the modulation equations. The equilibrium solutions of the modulation equations and the dynamic solutions of the ROM equation are investigated in the case of primary and principal parametric resonances of the first mode. Stability, bifurcations and frequency response curves of the nanobeam are investigated. Dynamic behaviors of the motion are shown in the form of time traces, phase portraits, Poincare sections and fast Fourier transforms. The results indicate rich dynamic behaviors such as Hopf bifurcations, periodic and quasiperiodic motions in both directly and indirectly excited modes illustrating the influence of modal interactions on the response.     

On Modal Interaction,Stability and Nonlinear Dynamics of a Model Two D.O.F. Mechanical System Performing Snap-Through Motion

Dynamics of a simple two degrees of freedom (d.o.f.) mechanical system is considered, to illustrate the phenomena of modal interaction. The system has a natural symmetry of shape and is subjected to symmetric loading. Two stable equilibrium configurations are separated by an unstable one, so that the model system can perform cross-well oscillations. Nonlinear statics and dynamics are considered, with the emphasis on detecting conditions for instability of symmetric configurations and analysis of bi-modal non-symmetric motions. Nonlinear local dynamics is analyzed by multiple scales method. Direct numerical integration of original equations of motions is carried out to validate analysis of modulation equations. In global dynamics (analysis of cross-well oscillations) Lyapunov exponents are used to estimate qualitatively a type of motion exhibited by the mechanical system. Modal interactions are demonstrated both in the local dynamics and for snap-through oscillations, including chaotic motions. This mechanical system may be looked upon as a lumped parameters model of continuous elastic structures (spherical segments, cylindrical panels, buckled plates, etc.). Analyses performed in the paper qualitatively describe complicated phenomena in local and global dynamics of original structures.     

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.     

The stability analysis of a slider-crank mechanism due to the existence of two-component parametric resonance

The objective of this paper is an analytical and numerical study of the dynamics and dynamic instability of a slider-crank mechanism with an inextensible elastic coupler. Special attention is given to the phenomena arising due to modal interactions produced by the existence of multi-component, specifically two-component, parametric resonance. Such modal couplings are very common in the bending-bending motions of fixed/ rotating beams. The two-component parametric resonance occurs when one of the natural frequencies of flexible parts of the mechanism is one-half or twice of the excitation frequency and simultaneously the sums or the differences among the internal frequencies are the same, or neighboring, as the frequency of excitation. The effects of two-component parametric resonance post on instability condition are also investigated. Resonance generated by more than two component modes are neglected due to its remote probability of occurrence in nature. The mechanics of the problem is Newtonian. Methods of analysis will consist of the dynamics of small deformations superimposed on the undeformed state. Without loss of generality and based on the Euler–Bernoulli beam theory, the coupled nonlinear equations of motion of a slider-crank mechanism with an inextensible flexible linkage are derived. The Newtons second law is used to obtain the boundary constraints at the piston end. Galerkins procedure was used to remove the dependence of spatial coordinates in the partial differential equations. The method of multiple time scales is applied to consider the steady state solutions and the occurrence of dynamic instability of the resulting multidegree-of-freedom dynamical system with time-periodic coefficients.     

Nonlinear dynamics of doubly curved shallow microshells

The nonlinear dynamical characteristics of a doubly curved shallow microshell are investigated thoroughly. A consistent nonlinear model for the microshell is developed on the basis of the modified couple stress theory (MCST) in an orthogonal curvilinear coordinate system. In particular, based on Donnell’s nonlinear theory, the expressions for the strain and the symmetric rotation gradient tensors are obtained in the framework of MCST, which are then used to derive the potential energy of the microshell. The analytical geometrically nonlinear equations of motion of the doubly microshell are obtained for in-plane displacements as well as the out-of-plane one. These equations of partial differential type are reduced to a large set of ordinary differential equations making use of a two-dimensional Galerkin scheme. Extensive numerical simulations are conducted to obtain the nonlinear resonant response of the system for various principal radii of curvature and to examine the effect of modal interactions and the length-scale parameter.     

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.     

A continuum theory for viscoelastic composite beams

Summary A dynamical continuum theory is developed for laminated composite beams. Starting with an assumed displacement- and temperature field, the one-dimensional approximate theory is consistently constructed within the frame of the three-dimensional theory of linear, nonisothermal, anisotropic, coupled viscoelasticity. Each constituent of the beam may possess different constant thickness and mechanical properties. All dynamic interactions between the adjacent constituents are included. Further, the effects of transverse shear and normal strains and rotatory inertia as well as those of cross-sectional distortion are all taken into account. The resulting equations consist of the macroscopic beam equations of motion and heat conduction, the kinematical relations, the initial and boundary conditions and the constitutive equations, and they govern the extensional, flexural and torsional motions of laminated composite beams. The special cases of constituents which made of either isotropic thermoviscoelastic or anisotropic thermoelastic materials are discussed briefly.Supported by the Office of Naval Research.With 1 figure     

Application of reconstitution multiple scale asymptotics for a two-to-one internal resonance in Magnetic Resonance Force Microscopy

In this paper we formulate an initial-boundary-value-problem describing the three-dimensional motion of a cantilever in a Magnetic Resonance Force Microscopy setup. The equations of motion are then reduced to a modal dynamical system using a Galerkin ansatz and the respective nonlinear forces are expanded to cubic order. The direct application of the asymptotic multiple scales method to the truncated quadratic modal system near a 2:1 internal resonance revealed conditions for periodic and quasiperiodic energy transfer between the transverse in-plane and out-of-plane modes of the MRFM cantilever. However, several discrepancies are found when comparing the asymptotic results to numerical simulations of the full nonlinear system. Therefore, we employ the reconstitution multiple scales method to a modal system incorporating both quadratic and cubic terms and derive an internal resonance bifurcation structure that includes multiple coexisting in-plane and out-of-plane solutions. This structure is verified and reveals a strong dependency on initial conditions in which orbital instabilities and complex out-of-plane non-stationary motions are found. The latter are investigated via numerical integration of the corresponding slowly-varying evolution equations which reveal that breakdown of quasiperiodic tori is associated with symmetry-breaking and emergence of irregular solutions with a dense spectral content.     

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.     

弹性——粘弹性复合结构模态理论

本文研究弹性-粘弹性复合结构动力学基本问题.复合结构动力学方程是一组微分积分方程,引入增广状态变量.将其变换为常规的状态方程.研究了状态方程特征解的性质.提出了"振荡模态"和"蠕变模态"慨念.给出了脉冲响应矩阵和传递函数矩阵,讨论了它们的特性.复合结构模态理论为其动特性和动响应分析提供理论依据.     

Systematic modeling of a chain of N-flexible link manipulators connected by revolute–prismatic joints using recursive Gibbs-Appell formulation

The main intent of this paper is to represent a systematic algorithm capable of deriving the equations of motion of N-flexible link manipulators with revolute–prismatic joints. The existence of the prismatic joints together with the revolute ones makes the derivation of governing equations difficult. Also, the variations of the flexible parts of the links, with respect to time cause the associated mode shapes of the links to vary instantaneously. So, to derive the kinematic and dynamic equations of motion for such a complex system, the recursive Gibbs-Appell formulation is applied. For a comprehensive and accurate modeling of the system, the coupling effects due to the simultaneous rotating and reciprocating motions of the flexible arms as well as the dynamic interactions between large movements and small deflections are also included. In this study, the links are modeled based on the Euler–Bernoulli beam theory and the assumed mode method. Also, the effects of gravity as well as the longitudinal, transversal and torsional vibrations have been considered in the formulations. Moreover, a recursive algorithm based on 3 ×  3 rotational matrices has been applied in order to derive the system’s dynamic equations of motion, symbolically and systematically. Finally, a numerical simulation has been performed by means of a developed computer program in order to demonstrate the ability of this algorithm in deriving and solving the equations of motion related to such systems.     

A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Müller-Liu theory and extend Müllers entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Müller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.Received: 21 March 2003, Accepted: 1 September 2003, Published online: 11 February 2004PACS: 05.70.Ln, 61.25.Hq, 61.30.-vCorrespondence to: I. Luca     

Parametric excitation of beams moving over supports

Studied in this work are the formulation of equations of motion and the response to parametric excitation of a uniform cantilever beam moving longitudinally over a single bilateral support. The equations of motion are generated by using assumed modes to discretize the beam, by regarding the support as a kinematic constraint, and by employing an alternate form of Kane's method that is particularly well suited to systems subject to constraints. Instability information is developed using the results of perturbation analysis for harmonic longitudinal motions of small amplitude and with Floquet theory for general periodic motions of any amplitude. Results demonstrate that definitive instability information can be obtained for a beam moving longitudinally over supports based on the frequencies of free transverse vibration of a beam that is longitudinally fixed.     

AN ACCURATE TWO-DIMENSIONAL THEORY OF VIBRATIONS OF ISOTROPIC,ELASTIC PLATES

An infinite system of two-dimensional equations of motion of isotropic elastic plates with edge and corner conditions are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions and a linear function of the thickness coordinate of the plate. The linear term in the expansion is to accommodate the in-plane displacements induced by the rotation of the plate normal in low-frequency flexural motions. A system of first-order equations of flexural motions and accompanying boundary conditions are extracted from the infinite system. It is shown that the present system of equations is equivalent to the Mindlin’s first-order equations, and the dispersion relation of straight-crested waves of the present theory is identical to that of the Mindlin’s without introducing any corrections. Reduction of present equations and boundary conditions to those of classical plate theories of flexural motions is also presented.     

Non-linear modal properties of non-shallow cables

A non-linear mechanical model of non-shallow linearly elastic suspended cables is employed to investigate the non-linear modal characteristics of the free planar motions. An asymptotic analysis of the equations of motion is carried out directly on the partial-differential equations overcoming the drawbacks of a discretization process. The direct asymptotic treatment delivers the approximation of the individual non-linear normal modes. General properties about the non-linearity of the in-plane modes of different type—geometric, elasto-static and elasto-dynamic—are unfolded. The spatial corrections to the considered linear mode shape caused by the quadratic geometric forces are investigated for modes belonging to the three mentioned classes. Moreover, the convergence of Galerkin reduced-order models is discussed and the influence of passive modes is highlighted.     

Blowup of solutions for the planar motions of rotating nonlinearly elastic rods

This paper treats the motion of flexible, extensible, shearable nonlinearly elastic rods, described by a geometrically exact theory, when they are confined to a plane rotating about a fixed axis at constant angular speed and when they are confined to a fixed plane with one end rotating at a constant angular speed about an axis perpendicular to the fixed plane. The paper gives restrictions on the constitutive equations and initial conditions that ensure that motions become unbounded at rapid rates as time becomes infinite. The analysis of these constitutive restrictions employs the theory of characteristics for single first-order semilinear partial differential equations.     

The Method of Averaging for Euler's Equations of Rigid Body Motion

We formulate the method of averaging for perturbations of Euler's equations of rotational motion. Euler's equations are three strongly nonlinear coupled differential equations that can be viewed as a three dimensional oscillator. The method of averaging is used to determine the long-term influence of perturbation terms on the motion by averaging about the nominal rigid body motion. The treatment is applicable to a large class of motions including precession with large nutation – it is not restricted to small motions about simple spins or nearly axi-symmetric bodies. Three examples are shown that demonstrate the accuracy of the method's predictions.     

ANALYSIS OF NONLINEAR DYNAMIC STABILITY OF LIQUID-CONVEYING PIPES

Nonlinearly dynamic stability of flexible liquid-conveying pipe in fluid structure interaction was analyzed by using modal disassembling technique . The effects of Poisson . Junction and Friction couplings in the wave-flowing-vibration system on the pipe dynamic stability were included in the analytical model constituted by four nonlinear differential equations . An analyzing example of cantilevered pipe was done to illustrate the dynamic stability characteristics of the pipe in the full coupling mechanisms , and the phase curves related to the first four modal motions were drawn . The results show that the dynamic stable characteristics of the pipe are very complicated in the complete coupling mechanisms, and the kinds of the singularity points corresponding to the various modal motions are different.