An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms.     

Stress recovery algorithm for reduced order models of mechanical systems in nonlinear dynamic operative conditions

Nonlinear forced response analyses of mechanical systems in the presence of contact interfaces are usually performed in built-in numerical codes on reduced order models (ROM). Most of the cases these derive from complex finite element (FE) models, resulting from the high accuracy the designers require in modeling and meshing the components in commercial FE software. In the technical literature several numerical methods are proposed for the identification of the nonlinear forced response in terms of a kinematic quantity (i.e. displacement, velocity and acceleration) associated either to the master degrees-of-freedom retained in the ROM, or to the slave ones after having expanded the reduced response through the reduction matrix. In fact, the displacement is the quantity usually adopted to monitor the nonlinear response, and to evaluate the effectiveness of a partially loose friction interface in damping vibrations, with respect to a linear case where no friction interfaces exist and no energy dissipation can take place. However, when a ROM is used the engineering quantities directly involved in the mechanical design, i.e. the strains and stresses, cannot be retrieved without a further data processing. Moreover, in the case of a strong nonlinear behavior of the mechanical joints, the distributions of the nonlinear strains and stresses over the structure is likely different than the one obtained as a superposition of linear mode shapes whose definition require a-priori assumptions on the boundary conditions at the contact interface. This means that the mentioned approximation cannot be used to predict the safety margins of a structure working in real (nonlinear) operative conditions. This paper addresses this topic and presents a novel stress recovery algorithm for the identification of the strains and stresses resulting from a nonlinear forced response analysis on a ROM. The algorithm is applied to a bladed disk with friction contacts at the shroud joint, which make the behavior of the blades nonlinear and non-predictable by means of standard linear analyses in commercial FE software.

     

Nonlinear oscillations of an elastic two-degrees-of-freedom pendulum

Nonlinear oscillations of the vertical plane swinging spring pendulum in the resonance case are studied (frequencies ratio regarding horizontal and vertical directions is equal to 1:2). Square and cubic terms of the Hamiltonian are taken into account. Novel normal form method, i.e., the so called invariant normalization is applied to solve the stated problem. Full system of integrals exhibits equations of the normal form, and solution for the pendulum coordinates is expressed via elementary functions. Frequencies of modes of oscillations are proportional to the first power of amplitude, and not to the second power as it is exhibited by one dimensional Duffing oscillator. Amplitudes of the modes are changed periodically, and energy from one mode is transited to energy of the second one, whereas the period of oscillations depends on the initial conditions. It is illustrated that asymptotic solution with small amplitudes approximates well numerical solution of the governing equations. In addition, an example of a periodic stable solution with constant amplitudes of the oscillation modes is given. Stability of this solution is proved.     

Nonlinear Modal Analysis of Structural Systems Using Multi-Mode Invariant Manifolds

In this paper, an invariant manifold approach is introduced for the generationof reduced-order models for nonlinear vibrations of multi-degrees-of-freedomsystems. In particular, the invariant manifold approach for defining andconstructing nonlinear normal modes of vibration is extended to the case ofmulti-mode manifolds. The dynamic models obtained from this technique capture the essential coupling between modes of interest, while avoiding coupling fromother modes. Such an approach is useful for modeling complex systemresponses, and is essential when internal resonances exist between modes.The basic theory and a general, constructive methodology for the method arepresented. It is then applied to two example problems, one analytical andthe other finite-element based. Numerical simulation results are obtainedfor the full model and various types of reduced-order models, including theusual projection onto a set of linear modes, and the invariant manifoldapproach developed herein. The results show that the method is capable ofaccurately representing the nonlinear system dynamics with relatively fewdegrees of freedom over a range of vibration amplitudes.     

非线性模态的分类和新的求解方法

引入不可分偶数维不变流形的概念来定义非线性模态．在此基础上，揭示出了一种新的模态——耦合非线性模态，并对实际系统中各种可能的模态进行了分类．这种分类可能是新的构筑非线性模态理论的框架．用此方法构造非线性模态，得到的模态振子具有范式的形式，形式最简、却能反映原系统在平衡点附近的主要动力学行为，且易于得到非线性频率及非线性稳定性等方面的信息．不仅适用于分析一般的多自由度系统，还可用于分析奇数维系统；不仅可构造内共振系统的非耦合模态，还可用于构造内共振耦合模态．从掌握的资料看，以前的方法还不能解决上述所有问题     

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

     

STUDY OF PARAMETRIC VIBRATION OF GYROSCOPIC CONTINUA BASED ON NONLINEAR NORMAL MODES AND A NUMERICAL ITERATIVE APPROACH

以脉动流输流管为例,利用非线性模态技术和一种数值迭代法研究陀螺连续体的非线性参数振动响应问题. 通过谐波平衡法将系统非线性非自治控制方程转化为拟自治方程,并在状态空间上利用不变流形法构造系统的非线性模态. 以对应自治系统的解为初值,采用一种数值迭代法来求解拟自治控制方程的模态系数,结果证明了该迭代法的快速收敛性. 在频域分析中得到了幅频响应和相空间上的不变流形,而在时域复模态分析中则发现了参激陀螺系统的正交相位差和行波振动现象.     

MILM hybrid identification method of fractional order neural-fuzzy Hammerstein model

Aiming at the difficult identification of fractional order Hammerstein nonlinear systems, including many identification parameters and coupling variables, unmeasurable intermediate variables, difficulty in estimating the fractional order, and low accuracy of identification algorithms, a multiple innovation Levenberg–Marquardt algorithm (MILM) hybrid identification method based on the fractional order neuro-fuzzy Hammerstein model is proposed. First, a fractional order discrete neuro-fuzzy Hammerstein system model is constructed; secondly, the neuro-fuzzy network structure and network parameters are determined based on fuzzy clustering, and the self-learning clustering algorithm is used to determine the antecedent parameters of the neuro-fuzzy network model; then the multiple innovation principle is combined with the Levenberg–Marquardt algorithm, and the MILM hybrid algorithm is used to estimate the linear module parameters and fractional order. Finally, the academic example of the fractional order Hammerstein nonlinear system and the example of a flexible manipulator are identified to prove the effectiveness of the proposed algorithm.

     

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.     

The Moment Finite-Element Scheme in Problems of Nonlinear Continuum Mechanics

The conceptual and theoretical fundamentals of the original moment finite-element scheme (MFES) developed to solve problems of nonlinear continuum mechanics are presented. Typical examples of model problems are given to illustrate the advantages of the MFES over other traditional finite-element schemes. The basic relationships for studying the nonlinear deformation and distortion of mechanical continuum systems are formulated in an invariant tensor form. Also, some mathematical algorithms specially developed to solve systems of nonlinear equations of high order are described. The numerical solutions obtained are proved reliable and rapidly converging to the exact solutions for a sufficient number of test problems. Results of strength, postbuckling stability, vibration, fracture, and high-speed influence analyses of real mechanical systems are illustrated.     

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.     

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

On nonlinear vibrations of a heavy mass point on a spring

We present a complete study of small nonlinear vibrations of a swinging spring with a nonlinear dependence of the spring tension on its elongation. We use the Hamiltonian normal form method. The Hamiltonian normal form profitably differs from the general normal form of differential equations, because it has an additional integral. To reduce the Hamiltonian to normal form, we use the invariant normalization method, which significantly reduces the computations. The normal form asymptotics are obtained by successively calculating the quadratures in the same way for both resonance and nonresonance cases. The solutions of Hamiltonian equations in normal form showed that the periodic change of vibrations from vertical to horizontal modes and vice versa occurs only in the case of 1:1 and 2:1 resonances. In the case of 2:1 resonance, this effect manifests itself in the quadratic terms of the equation, and in the case of 1:1 resonance, it manifests itself if the cubic terms are taken into account. In all other cases, both in the case of resonance and without any resonance, the vibrations occur at two constant frequencies, which slightly differ from the linear approximation frequencies. In the case of 2:1 resonance, we found the maximum frequency detuning at which the effect of the energy pumping from one vibration mode to another disappears. 1:1 resonance is physically possible only for a spring with a negative cubic additional term in the strain law.     

Chaotic response of a large deflection beam and effect of the second order mode

This study intends to investigate the dynamic behavior of a nonlinear elastic beam of large deflection. Using the Galerkin principle, the dynamic nonlinear governing equations are derived based on the single and double mode methods. Two different kinds of nonlinear dynamic equations are obtained with the variation of the dimension and loading parameters. The chaotic critical conditions are given by Melnikov function method for the single mode model. The chaotic motion is investigated and the comparison between single and double mode models is carried out. The results show that the single mode method usually used may lead to incorrect conclusions in some conditions, and instead the double mode or higher order mode method should be used. Finally, the applicable condition of the single mode method is analyzed.     

Second-harmonic generation of two-dimensional elastic wave propagation in an infinite layered structure with nonlinear spring-type interfaces

The two-dimensional elastic wave propagation in an infinite layered structure with nonlinear interlayer interfaces is analyzed theoretically to investigate the second-harmonic generation due to interfacial nonlinearity. The structure consists of identical isotropic linear elastic layers that are bonded to each other by spring-type interfaces possessing identical linear normal and shear stiffnesses but different quadratic nonlinearity parameters. Explicit analytical expressions are derived for the second-harmonic amplitudes when a single monochromatic Bloch mode propagates in the structure in arbitrary directions by applying the transfer-matrix approach and the Bloch theorem to the governing equations linearized by a perturbation method. The second-harmonic generation by a single nonlinear interface and by multiple consecutive nonlinear interfaces are shown to be profoundly influenced by the band structure of the layered structure, the fundamental Bloch wave mode, and its propagation direction. In particular, the second harmonics generated at multiple consecutive interfaces are found to grow cumulatively with the propagation distance when the phase matching occurs between the Bloch modes at the fundamental and double frequencies.     

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.     

有间隙折叠舵面的振动实验与非线性建模研究

针对折叠舵面内、外舵铰接处存在的间隙对地面振动响应的影响及间隙处的非线性建模方法展开研究.消除间隙,利用锤击法对线性折叠舵面进行模态实验,得到了前五阶模态参数;打开间隙,进行振动台扫频基础激励,实验结果表明间隙的存在会使结构的动力学响应产生非线性现象,如正反向扫描差异、跳跃、多谐波及频率漂移.非线性的影响主要体现在一阶弯曲模态上,激励量级的增大和间隙的减小均会使基频增大,且逐渐趋向于无间隙的结果,但对第二阶扭转模态的影响与第一阶相比较小.建立了折叠舵面的有限元模型. 提出了一种适用于具有集中非线性的折叠机构的模型缩减方法,并对舵面进行了模态缩减.根据Hertz接触理论,用具有线性和3/2次刚度组合形式的非线性扭转弹簧来模拟铰接处的间隙和接触.通过比较锤击实验与数值计算得到的前四阶频率和振型对模型的线性部分进行验证.通过Bathe两子步隐式复合算法计算基础激励下非线性结构的动力学响应,得到的传递函数可以模拟实验中出现的频率变化特征,验证了连接处非线性建模方法的合理性.      

Submodels of model of nonlinear diffusion with non-stationary absorption

We study the model, describing a nonlinear diffusion process (or a heat propagation process) in an inhomogeneous medium with non-stationary absorption (or source). We found tree submodels of the original model of the nonlinear diffusion process (or the heat propagation process), having different symmetry properties. We found all invariant submodels. All essentially distinct invariant solutions describing these invariant submodels are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. For example, we obtained the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with two fixed "black holes", and the invariant solution describing the nonlinear diffusion process (or the heat distribution process) with the fixed "black hole" and the moving "black hole". The presence of the arbitrary constants in the integral equations, that determine these solutions provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original model of the nonlinear diffusion process (or the heat distribution process). For the received invariant submodels we are studied diffusion processes (or heat distribution process) for which at the initial moment of the time at a fixed point are specified or a concentration (a temperature) and its gradient, or a concentration (a temperature) and its rate of change. Solving of boundary value problems describing these processes are reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. The obtained results can be used to study the diffusion of substances, diffusion of conduction electrons and other particles, diffusion of physical fields, propagation of heat in inhomogeneous medium.     

Experimental analysis of nonlinear resonances in piezoelectric plates with geometric nonlinearities

Piezoelectric devices with integrated actuation and sensing capabilities are often used for the development of electromechanical systems. The present paper addresses experimentally the nonlinear dynamics of a fully integrated circular piezoelectric thin structure, with piezoelectric patches used for actuation and other for sensing. A phase-locked loop control system is used to measure the resonant periodic response of the system under harmonic forcing, in both its stable and unstable parts. The single-mode response around a symmetric resonance as well as the coupled response around an asymmetric resonance, involving two companion modes in 1:1 internal resonance, is accurately measured. For the latter, a particular location of the patches and additional signal processing is proposed to spatially discriminate the response of each companion mode. In addition to a hardening behavior associated with geometric nonlinearities of the plate, a softening behavior predominant at low actuation amplitudes is observed, resulting from the material piezoelectric nonlinearities.

     

Analysis of guided waves with a nonlinear boundary condition caused by internal resonance using the method of multiple scales

We theoretically investigated the cumulative nonlinear guided waves caused by internal resonance, using the method of multiple scales (MMS), which can construct better approximations to the solutions of perturbation problems. In this study, we consider nonlinearity only on the boundary instead of material nonlinearity or geometric nonlinearity. We showed nonlinear effects on the amplitudes of a lower mode and a higher mode depending on the propagation length. Also, we examined effects of wavenumber detuning from a phase matching condition of the two modes. If the wavenumber detuning is exactly equal to zero, the mechanical energy of the lower mode is transferred through nonlinear coupling to the energy of the higher mode, unilaterally. However, if a wavenumber detuning is not equal to zero, amplitude of the two modes change in a cyclic fashion during wave propagation. The amount of this amplitude variation and its cycle length are determined by the eigenfunctions of the two modes, the nonlinear parameter and the wavenumber detuning.