SINGULAR ANALYSIS OF BIFURCATION OF NONLINEAR NORMAL MODES FOR A CLASS OF SYSTEMS WITH DUAL INTERNAL RESONANCES

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Singular Characteristics of Nonlinear Normal Modes in a Two Degrees of Freedom Asymmetric Systems with Cubic Nonlinearities

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｍｏｄｅｒｎａｎａｌｙｓｉｓａｎｄｍｅｔｈｏｄｓｆｏｒｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓｈａｖｅｇｒｅａｔｌｙｐｒｏｍｏｔｅｄｔｈｅｄｅｖｅｌｏｐｍｅｎｔｉｎｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅ.ＴｈｅｓｅｉｎｃｌｕｄｅＬ＿Ｓｒｅｄｕｃｅ[1],ｓｉｎｇｕｌａｒｉｔｙｔｈｅｏｒｙ[2 ],ｐｅｒｔｕｒｂａｔｉｏｎｔｅｃｈｎｉｑｕｅ[3 ],Ｍｅｌｎｉｋｏｖｆｕｎｃｔｉｏｎ[4 ],Ｃ＿Ｌｍｅｔｈｏｄ[5 ]ａｎｄｃｅｎｔｅｒｍａｎｉｆｏｌｄ[6],ｅｔｃ .Ｈｏｗｅｖｅｒ,ｔｈｅｃｏｕｐｌｉｎｇｂｅ…     

A very complicated structure of resonances in a nonlinear system with cyclic symmetry: Nonlinear forced localization

The fundamental and subharmonic resonances of a nonlinear cyclic assembly are examined using the asymptotic method of multiple-scales. The system consists of a number of identical cantilever beams coupled by means of weak linear stiffnesses. Assuming beam inextensionality, geometric nonlinearities arise due to longitudinal inertia and the nonlinear relation between beam curvature and transverse displacement. The governing nonlinear partial differential equations are discretized by a Galerkin procedure and the resulting set of coupled ordinary differential equations is solved using an asymptotic analysis. The unforced assembly is known to possess localized nonlinear normal modes, which give rise to a very complicated topological structure of fundamental and subharmonic response curves. In contrast to the linear system which exhibits as many forced resonances as its number of degrees of freedom, the nonlinear system is found to possess a number of additional resonance branches which have no counterparts in linear theory. Some of the additional resonances are spatially localized, corresponding to motions of only a small subset of periodic elements. The analytical results are verified by numerical Poincaré maps, and the forced localization features of the nonlinear assembly are demonstrated by considering its response to impulsive excitations.     

Nonlinear normal modes and their superposition in a two degrees of freedom asymmetric system with cubic nonlinearities

This paper investigates nonlinear normal modes and their superposition in a two degrees of freedom asymmetric system with cubic nonlinearities for all nonsingular conditions, based on the invariant subspace in nonlinear normal modes for the nonlinear equations of motion. The focus of attention is to consider relation between the validity of superposition and the static bifurcation of modal dynamics. The numerical results show that the validity has something to do not only with its local restriction, but also with the static bifurcation of modal dynamics. Project Supported by the National Natural Science Foundation and PSF of China     

Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators

Redistribution of energy in a highly asymmetric system consisting ofcoupled linear and highly nonlinear damped oscillators isinvestigated. Special attention is paid to the excitation of a nonlinearnormal mode while the energy is initially stored in other modes of thesystem. The transition proceeds via the mechanism of subharmonicresonance which is possible because of the strong nonlinearity of thesystem. The conditions of the energy transition to NNM being effectiveare revealed and guidelines to design such a systems are formulatedin detail.     

Karman型正交异性矩形薄板非线性弯曲的统一求解方法

本文对Karman型四边支承正交异性薄板在5种不同边界条件下的几何非线性弯曲进行了统一分析。所设的位移函数均为梁振动函数。它们精确地满足边界条件，利用Galerkin方法和位移函数的正交属性，转换控制方程为非线性代数方程。用“稳定化双共轭梯度法”求解稀疏矩阵线性方程组以及“可调节参数的修正迭代法”求解非线性代数方程组，最后给出了相应的数值结果。     

The construction of non-linear normal modes for systems with internal resonance

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.     

Stress and strain fields in rotating elastic/plastic annular disks of pressure-dependent material

The Drucker–Prager yield criterion is used in conjunction with its associated flow rule to find the elastic/plastic stress and strain distributions within the rotating annular disks under plane stress conditions. The main distinguished feature of the model, as compared to typical models used for analysis of disks, is that the material is plastically compressible. Using an approach proposed elsewhere, the solution for strain rates is reduced to one nonlinear ordinary differential equation and two linear ordinary differential equations. These equations can be solved one by one, which significantly simplifies the numerical treatment and increases the accuracy of solution.     

弹塑性接触问题的非光滑非线性方程组方法

将求解三维弹性摩擦接触问题的非光滑非线性方程组方法推广到弹塑性(Mises材料)情形，提出了两种应用方法：一种是将非光滑非线性方程组方法和求解弹塑性问题常用的Newton—Raphson迭代方法结合起来；另一种是将问题写成统一的非光滑非线性方程组，直接求解。数值算例验证了两种方法的有效性，并进行了结果比较。     

Principal parametric and three-to-one internal resonances of flexible beams undergoing a large linear motion

A set of nonlinear differential equations is established by using Kane‘s method for the planar oscillation of flexible beams undergoing a large linear motion. In the case of a simply supported slender beam under certain average acceleration of base, the second natural frequency of the beam may approximate the tripled first one so that the condition of 3 : 1 internal resonance of the beam holds true. The method of multiple scales is used to solve directly the nonlinear differential equations and to derive a set of nonlinear modulation equations for the principal parametric resonance of the first mode combined with 3 : 1 internal resonance between the first two modes. Then, the modulation equations are numerically solved to obtain the steady-state response and the stability condition of the beam. The abundant nonlinear dynamic behaviors, such as various types of local bifurcations and chaos that do not appear for linear models, can be observed in the case studies. For a Hopf bifurcation,the 4-dimensional modulation equations are reduced onto the central manifold and the type of Hopf bifurcation is determined. As usual, a limit cycle may undergo a series of period-doubling bifurcations and become a chaotic oscillation at last.     

NONLINEAR DYNAMICS AND SYNCHRONIZATION OF TWO COUPLED PIPES CONVEYING PULSATING FLUID

In this paper, the nonlinear dynamical behavior of two coupled pipes conveying pulsating fluid is studied. The connection between the two pipes is considered as a distributed linear spring. Based on this consideration, the equations of motion of the coupled two-pipe system are obtained. The two coupled nonlinear partial differential equations, discretized using the fourth- order Galerkin method, are solved by a fourth-order Runge-Kutta integration algorithm. Results show that the connection stiffness has a significant effect on the dynamical behavior of the coupled system. It is found that for some parameter values the motion types of the two pipes might be synchronous.     

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.     

一维非线性黏弹体和黏塑性体中的最优应变路径

得出了一维非线性黏弹体和黏塑性体中最优应变路径方程，并揭示了它们的某些特性，如：（１）当非线性黏弹本构方程中的黏性部分与应变的关系具有上凸形式时，相应的最优应变路径具有下凸性质；（２）对于过应力和Ｂｏｄｎｅｒ－Ｐａｒｔｏｎ黏塑性体，它们的最优应变路径是塑性应变为线性形式，即塑性应变率为常数，而弹－黏塑性体的最优应变路径则不同。     

A hybrid approach for nonlinear computational aeroacoustics predictions

In many aeroacoustics applications involving nonlinear waves and obstructions in the far-field, approaches based on the classical acoustic analogy theory or the linearised Euler equations are unable to fully characterise the acoustic field. Therefore, computational aeroacoustics hybrid methods that incorporate nonlinear wave propagation have to be constructed. In this study, a hybrid approach coupling Navier–Stokes equations in the acoustic source region with nonlinear Euler equations in the acoustic propagation region is introduced and tested. The full Navier–Stokes equations are solved in the source region to identify the acoustic sources. The flow variables of interest are then transferred from the source region to the acoustic propagation region, where the full nonlinear Euler equations with source terms are solved. The transition between the two regions is made through a buffer zone where the flow variables are penalised via a source term added to the Euler equations. Tests were conducted on simple acoustic and vorticity disturbances, two-dimensional jets (Mach 0.9 and 2), and a three-dimensional jet (Mach 1.5), impinging on a wall. The method is proven to be effective and accurate in predicting sound pressure levels associated with the propagation of linear and nonlinear waves in the near- and far-field regions.     

圆柱壳的非线性自由振动分析

本文分析了各向同性封闭圆柱壳的非线性自由振动。文中采用经典的非线性弹性力学方法推导了圆柱壳的大振幅运动方程,这些方程的静态形式与冯·卡门的板理论方程具有同样的精度。文中讨论了四种基本振动模态,并且还以数学公式的形式给出了一般的最终结果,一些例子以曲线给出结果,并进行了比较。结果还表明线性振动可以作为非线性振动的一种特例。     

Nonlinear Normal Modes of a Parametrically Excited Cantilever Beam

We investigate theoretically thenonlinear normal modes of a vertical cantilever beam excited by aprincipal parametric resonance. We apply directly the method ofmultiple scales to the governing nonlinear nonautonomousintegral-partial-differential equation and associated boundary conditions.In the absence of damping, it is shown that the system has nonlinear normal modes, as defined by Rosenberg, even in the presence of the parametric excitation.We calculate the spatial correction to the linear mode shapedue to the effects of the inertia and curvature nonlinearities andthe parametric excitation. We compare the result obtained withthe direct approach with that obtained using a single-mode Galerkindiscretization.The deviation between the two predictions increases as the oscillationamplitude increases.     

高阶剪切变形理论下三边夹紧一边铰支复合材料层板的几何非线性分析

首先用虚位移原理推导出以位移形式表达的Reddy型高阶剪变形理论复合材料层板的非线性控制方程及相应的边界条件。选定的五个位移函数均满足三边夹紧一边铰支边界条件，用Galerkin方法把无量纲化之后的控制方程转化为一组非线性代数方程组，用线性化的方法和可调节参数的修正迭代法求解这组方程。最后求出了不同复合材料的挠度和弯矩值。     

Nonlinear stability of truncated shallow spherical shell with variable thickness under uniformly distributed load

I.Intr0ducti0nTheclampedtruncatedshailowsphericalshellwithanondeformablerigidbodyatthecenterisoftenusedinstructureandelasticcomponentofprecisioninstrument.Inthebuildingengineering,wemustpreventtheshelllosingstability.Butintheelasticcomponent0fprecisionins…     

Component Mode Synthesis Using Nonlinear Normal Modes

This paper describes a methodology for developing reduced-order dynamic models of structural systems that are composed of an assembly of nonlinear component structures. The approach is a nonlinear extension of the fixed-interface component mode synthesis (CMS) technique developed for linear structures by Hurty and modified by Craig and Bampton. Specifically, the case of nonlinear substructures is handled by using fixed-interface nonlinear normal modes (NNMs). These normal modes are constructed for the various substructures using an invariant manifold approach, and are then coupled through the traditional linear constraint modes (i.e., the static deformation shapes produced by unit interface displacements). A class of systems is used to demonstrate the concept and show the effectiveness of the proposed procedure. Simulation results show that the reduced-order model (ROM) obtained from the proposed procedure outperforms the ROM obtained from the classical fixed-interface linear CMS approach as applied to a nonlinear structure. The proposed method is readily applicable to large-scale nonlinear structural systems that are based on finite-element models.     

On averaged and normal form equations

The method of ayeraging has long been an important method for the analysis of nonlinear systems. One method using normal forms has more recently become a tool in applied mathematics. Both methods convert systems of ordinary differential equations to simpler equations by using a sequence of change of variables. This study can be interpreted as a local analysis in that it treats equations that are nonlinear perturbations of linear equations. We show that the two methods are essentially equivalent.This work has been supported by the National Science Foundation, Grant No. NSF/MEA 8310966. Deceased.