The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N-1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.     

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.      

Vibration absorbers for a rotating flexible structure with cyclic symmetry: nonlinear path design

This paper analytically investigates the nonlinear dynamics of order-tuned vibration absorbers applied to cyclic rotating flexible structures under traveling wave (TW) engine-order excitation. The primary cyclic structure is assumed to be governed by linear vibrations and the nonlinear absorber response arises from large amplitude kinematic effects. These dynamics are captured by a lumped-parameter model that consists of N blades with one blade mode and one absorber per blade, which are arranged with cyclic symmetry on a rotating disk. The governing equations of motion are formulated for arbitrary absorber paths to allow investigation of the absorber path design for nonlinear response. This paper extends previous work by the authors, which considered the linearized blade and absorber dynamics of a similar system. Several intriguing features of the dynamics were uncovered, most notably the existence of an absorber tuning range that avoids resonance at any rotation speed. Of particular interest is the existence and stability of the steady-state TW response to TW excitation, as experienced in turbomachinery, and how these are affected by selection of the absorber paths, which fix the linear and nonlinear tuning characteristics. It is shown that the TW response, which is unique for the linearized system, also exists for the weakly nonlinear model and can be captured by an equivalent two degree of freedom model obtained using the symmetry of the excitation and system response. The forced response exhibits the usual characteristics of a weakly nonlinear system, specifically, bistability and the attendant hysteresis near resonance. More significantly, it does not experience any additional instabilities associated with the symmetry. That is, the desired TW response is robust to nonlinear effects in the absorber, which allows use of the simple equivalent model for selection of absorber tuning parameters. For good performance and robustness, the linear absorber tuning should be in the “no-resonance zone” described by the linear theory and the absorber paths should have a slightly softening nonlinear characteristic.     

Energy Flow in Formally Gradient Partial Differential Equations on Unbounded Domains

As an example of an extended, formally gradient dynamical system, we consider the damped hyperbolic equation u _tt+u _t=u+F(x, u) in R ^N , where F is a locally Lipschitz nonlinearity. Using local energy estimates, we study the semiflow defined by this equation in the uniformly local energy space H¹ _ul(R ^N )×L² _ul(R ^N ). If N2, we show in particular that there exist no periodic orbits, except for equilibria, and we give a lower bound on the time needed for a bounded trajectory to return in a small neighborhood of the initial point. We also prove that any nonequilibrium point has a neighborhood which is never visited on average by the trajectories of the system, and we conclude that any bounded trajectory converges on average to the set of equilibria. Some counter-examples are constructed, which show that these results cannot be extended to higher space dimensions.     

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.     

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.

     

Minimum distance and optimal transformations on SO(N)

The group of special (or proper) orthogonal matrices, SO(N), is used throughout engineering mechanics in the analysis and representation of mechanical systems. In this paper, a solution is presented for the optimal transformation between two elements of SO(N). The transformation is assumed to occur during a specified finite time, and a cost function that penalizes the transformation rates is utilized. The optimal transformation is found as a constant-rate rotation in each of the principal planes relating the two elements. Although the kinematics of SO(N) are nonlinear and governed by Poisson’s equation, the solution is found to be a linear function of the generalized principal angles. This is made possible by the extension of principal-rotation kinematics from three-dimensional rotations to the general SO(N) group. This extension relates the N-dimensional angular velocity to the derivatives of the principal angles. The cost of the optimal transformation, the square root of the sum of the principal angles squared, also provides a useful measure for the angular distance between two elements of SO(N).     

A time domain collocation method for obtaining the third superharmonic solutions to the Duffing oscillator

In this study, a simple time domain collocation method (TDC) is applied to investigate the third superharmonic solutions of the Duffing oscillator. Upon using the proposed scheme, the multivaluedness, jump phenomenon, and transitional region of the third superharmonic response are explored. The amplitude frequency response curves for various values of damping, nonlinearity, and external force are obtained and compared. In addition, instead of collocating at N points so that the resulting nonlinear algebraic system is well determined, we extend the time domain collocation method to a new version by collocating at M>N points. The resulting over determined system is solved by the least square method. The extended time domain collocation method can significantly relieve the nonphysical solution phenomenon, which may be severe in the time domain collocation method, and its equivalent high dimensional harmonic balance method. Finally, numerical examples confirm the simplicity, efficiency, and accuracy of the proposed scheme.     

平稳高斯激励下线性结构随机振动分析的辅助简谐激励广义法

相比于时域法,频域法是更为高效、易行的随机振动分析方法,但对于平稳激励下的随机振动分析,现有频域方法常需振型截断或功率谱矩阵分解,将会影响计算精度和效率.为此,本文在频域法的框架下,针对平稳高斯激励下线性结构的随机振动分析提出了一种精确且高效的辅助简谐激励广义法.首先,引入广义脉冲响应函数和广义频响函数的概念,推导了与...     

Experiments on a beam-rigid body structure repetitively impacting a rod

This paper investigates the experimental dynamics of a beam structure that supports an attached rigid body and that can impact a comparatively compliant base structure. The problem area is motivated by impact phenomena that are observed in certain structures internal to nuclear reactors. The assembly is subjected to base excitation at specified frequency and acceleration, and the resulting displacement and velocity time histories are recorded and used to obtain spectra, phase diagrams, and Poincaré sections. The measurements validate simulation results obtained by using a constraint and modal mapping method based on the two sets of modes when the structure is in-contact, and when it is not-in-contact. Generalized coordinates are mapped across the impact discontinuities in the modal representation. The forced response simulation predicts the test specimen’s response over a range of excitation frequencies. The specimens are fabricated as single integral structures from acrylnitrile butadene styrene plastic through rapid prototyping technology in order to eliminate the undesirable dissipation and flexibility arising from joints and connections. The experimental system can exhibit complex response characteristics, and the influences on complexity of deadband clearance and of asymmetry in the point of impact are examined in the experiments.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

Nonlinear dynamic behavior of a bio-inspired embedded X-shaped vibration isolation system

To improve the vibration isolation performance and bandwidth, loading capacity and supporting stability of passive vibration isolation system by utilizing nonlinearity, a bio-inspired embedded X-shaped vibration isolation (BIE-XVI) structure is proposed considering muscle/tendon contractile functions, joint rotational friction and connecting rod mass simultaneously. Furthermore, the dynamic model with pure linear elements and geometric relationship are established and the nonlinear variation properties are investigated. The effects of the key parameters of the BIE-XVI structure on frequency response characteristics and vibration isolation range are analyzed thoroughly by incremental harmonic balance method in various working conditions. From the parametric investigations, it can be found that the sensitivities of the nonlinear resonance properties are markedly different with respect to the different structure parameters. For longer rod length, larger assembly angle and higher stiffnesses, the hardening nonlinearity is weakened, but the resonance peak does not necessarily decrease. Besides, the softening nonlinearity and hardening nonlinearity can be interconverted with changing isolated mass and excitation amplitude. The BIE-XVI structure can widen the isolation frequency range and reduce the resonance peak to improve the vibration isolation properties by adjusting/designing the structural parameters, which could realize quasi-zero-stiffness property for vibration isolation.

     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

参数和直接激励下非线性压电俘能器性能分析的多尺度法

考虑几何非线性、阻尼非线性和梁的轴向不可伸长条件,利用Hamilton变分原理,建立了参数激励和直接激励下压电俘能器的非线性力电耦合的运动微分方程;利用Galerkin法,将所建立的动力学偏微分方程降阶为力电耦合的Mathieu-Duffing型方程;采用多尺度法获得了梁的位移和输出电压的解析表达式,给出了解的稳定性条件;利用解析表达式研究了单独参数激励以及参数激励和直接激励共同作用下阻尼系数对压电俘能器性能的影响。结果表明,在参数激励情况下,线性阻尼会显著影响超临界分岔点的位置,非线性二次阻尼不会影响超临界分岔点的位置。参数激励和直接激励的结合可以作为提升压电能量俘获器性能的解决方案。     

Existence of planar flame fronts in convective-diffusive periodic media

We prove the existence of planar travelling wave solutions in a reaction-diffusion-convection equation with combustion nonlinearity and self-adjoint linear part in R ⁿ, n1. The linear part involves diffusion-convection terms and periodic coefficients. These travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media. We use the method of continuation, spectral theory, and the maximum principle. Uniqueness and monotonicity properties of solutions follow from a previous paper. These properties are essential to overcoming the lack of compactness and the degeneracy in the problem.     

Numerical calculation of nonlinear normal modes in structural systems

This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to “artificially destabilize” the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient ε of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for ε = 0 into the manifold existing quasi-statically for instantaneous values of ε. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).     

Solitary waves for power-law regularized long-wave equation and <Emphasis Type="Italic">R</Emphasis>(<Emphasis Type="Italic">m</Emphasis>,<Emphasis Type="Italic">n</Emphasis>) equation

This paper integrates the regularized long-wave equation with power-law nonlinearity using the solitary-wave ansatz. A few of the conserved quantities are calculated by using the 1-soliton solution. This technique is then extended to obtain the solitary-wave solution of the R(m,n) equation and a conserved quantity is also calculated for this generalized equation.     

The Multiconfiguration Equations for Atoms and Molecules: Charge Quantization and Existence of Solutions

We prove the existence of ground-state solutions for the multiconfiguration self-consistent field equations for atoms and molecules whenever the total nuclear charge Z exceeds N–1, where N is the number of electrons. Moreover, we show that for arbitrary values of Z and N the scattering charge, i.e., the asymptotic amount of charge lost by an energy-minimizing sequence, is integer-quantized. Our analysis applies to the MC equations of arbitrary rank. As special cases we recover, in a new and unified way, the existence theorems of Zhislin [Zh60] for the N-body Schrödinger equation (infinite rank MC) and of Lieb & Simon [LS77] for the Hartree-Fock equations (rank-N MC). Our approach is a direct study of an invariant, orbital-free formulation in N-body space of the underlying variational principle. Proofs involve (i) the geometric N-body localization methods for the linear Schrödinger equation first introduced by Enss [En77] (and developed in [Sim77, Sig82]), which can be adapted to become powerful tools in nonlinear many-body theory as well, (ii) weak convergence methods from the theory of nonlinear partial differential equations, (iii) careful analysis of the structure of the one- and two-body density matrices of the bound and scattering fragments delivered by geometric localization, which allows us to overcome the fact that the rank of the fragments is not reduced by localization.