Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method

This paper addresses the large-amplitude free vibration of simplysupported Timoshenko beams with immovable ends. Various nonlineareffects are taken into account in the present formulation and thegoverning differential equations are established based on theHamilton Principle. The differential quadrature method (DQM) isemployed to solve the nonlinear differential equations. Theeffects of nonlinear terms on the frequency of the Timoshenkobeams are discussed in detail. Comparison is made with otheravailable results of the Bernoulli–Euler beams and Timoshenkobeams. It is concluded that the nonlinear term of the axial forceis the dominant factor in the nonlinear vibration of Timoshenkobeams and the nonlinear shear deformation term cannot be neglectedfor short beams, especially for large-amplitude vibrations.     

非线性系统参数辩识的一种频域模型

本文基于对非线性系统的可分离性假设，将非线性弹性力和阻尼力分别分解为物理坐标下各点间相对位移和相对速度的幂级数函数，导出了一般多自由度非线性系统在恒幅激励下的广义频率响应函数与输入输出之间的迭代关系式，提出了非线性系统中基本线性部分的概念，进而了一种在实验条件下的系统物理参数辩识方法。     

A Simple Nonlinear Thermodynamic Theory of Arbitrary Elastic Beams

A ‘classical’ theory of beams (i.e., a theory in which the basic kinetic variables are a stress resultant and a stress couple) undergoing elastic, thermodynamic processes is developed by first deriving exact beamlike (one-dimensional) equations of motion and a beamlike Second Law (Clausius–Duhem inequality) by descent from three-dimensions. Then what may be considered as the three basic assumptions of a classical theory are introduced: an assumed form of the First Law (conservation of energy), a relaxed form of the Second Law, and a general form of the constitutive relations. Throughout, detailed specification of geometry, kinematics, or constitution is minimized. It is shown how the kinematic Kirchhoff hypothesis may be avoided by first introducing a mixed-energy density and then imposing a logically more satisfying constitutive Kirchhoff hypothesis. Mathematics Subject Classifications (2000) 74A15, 74B20, 74K10     

Nonlinear System Identification of Multi-Degree-of-Freedom Systems

A nonlinear system identification methodology based on theprinciple of harmonic balance is extended tomulti-degree-of-freedom systems. The methodology, called HarmonicBalance Nonlinearity IDentification (HBNID), is then used toidentify two theoretical two-degree-of-freedom models and anexperimental single-degree-of freedom system. The three modelsand experiments deal with self-excited motions of afluid-structure system with a subcritical Hopf bifurcation. Theperformance of HBNID in capturing the stable and unstable limitcycles in the global bifurcation behavior of these systems is alsostudied. It is found that if the model structure is well known,HBNID performs well in capturing the unknown parameters. If themodel structure is not well known, however, HBNID captures thestable limit cycle but not the unstable limit cycle.     

Distinguishing Periodic and Chaotic Time Series Obtained from an Experimental Nonlinear Pendulum

The experimental analysis of nonlinear dynamical systems furnishes ascalar sequence of measurements, which may be analyzed using state spacereconstruction and other techniques related to nonlinear analysis. Thenoise contamination is unavoidable in cases of data acquisition and,therefore, it is important to recognize techniques that can be employedfor a correct identification of chaos. The present contributiondiscusses the experimental analysis of a nonlinear pendulum, consideringstate space reconstruction, frequency domain analysis and thedetermination of dynamical invariants, Lyapunov exponents and attractordimension. A procedure to construct Poincaré map of the signal ispresented. The analyses of periodic and chaotic motions are carried outin order to establish a difference between them. Results show that it ispossible to distinguish periodic and chaotic time series obtained froman experimental set up employing proper procedures even though noisesuppression is not contemplated.     

Enhanced Nonlinear Dynamics for Accurate Identification of Stiffness Loss in a Thermo-Shielding Panel

The dynamics of a panel forced by transverse loads and undergoing limit cycle oscillations and chaos is investigated. The nonlinear von Karman plate theory is used to obtain a model for healthy and damaged panels. Damage is modeled by a loss of stiffness in a portion of the plate. The presence of low levels of damage is identified by using an external nonlinear excitation and analyzing the attractor of the resulting dynamics in state space. Most of the current studies of such problems are based on linear theories and linear structures. In contrast, the results presented are obtained by using and enhancing nonlinear and chaotic dynamics, and have the advantage of an increased accuracy in detecting damage and monitoring structural health.An earlier version of this paper was presented at the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm Springs, California, April 2004.     

用非线性模态综合法模拟碰撞接触区应力

本文从振动分析的观点出发,研究了线弹性系统的碰撞问题.文中借助有限元方法确定参与碰撞之系统的动力学特性,然后应用非线性模态综合法对碰撞过程进行模拟计算,得到了接触面皮下微深处的应力分布及其随时间变化的规律.文中指出:只要有碰撞存在,接触表面的挤压应力就一定能够超过材料的线弹性极限而出现塑性变形或破坏.     

Nonlinear System Identification of Systems with Periodic Limit-Cycle Response

A nonlinear system identification methodology based on the principle of harmonic balance and bifurcation theory techniques like center manifold analysis and normal form reduction, is presented for multi-degree-of-freedom systems. The methodology, called Bifurcation Theory System IDentification, (BiTSID), is a general procedure for any nonlinear system that exhibits periodic limit cycle response and can be used to capture the bifurcation behavior of the nonlinear systems. The BiTSID methodology is demonstrated on an experimental system single-degree-of-freedom system that deals with self-excited motions of a fluid-structure system with a sub-critical Hopf bifurcation. It is shown that BiTSID performs excellently in capturing the stable and unstable limit cycles within the experimental regime. Its performance outside the experimental regime is also studied. The application of BiTSID to experimental multi-degree-of-freedom systems has also been very successful. However in this study only the results of the single-degree-of-freedom system are presented.     

橡胶隔振系统非线性振动参数识别与特性研究

建立单自由度非线性系统基础激励正弦扫描试验的数学模型,并针对多项式非线性阻尼与多项式非线性弹簧并联的橡胶隔振系统求解了数学模型.进而在此框架下通过正弦扫描振动试验,采用共振峰值法识别了橡胶隔振器的非线性阻尼与刚度系数,研究了不同振动量级下小型橡胶隔振器的振动特性.     

钢筋混凝土T型梁在不同荷载(裂纹)下的试验模态分析及非线性外推

混凝土结构在大荷载作用下出现裂纹,其模态参数呈非线性规律.本文采用分段线性假设进行简化,即认为结构在不同级的荷载下对应有局部的线性特性.对钢筋混凝土T型简支梁在若干不同等级纯力加载并产生相应裂纹的情况下,分别进行试验模态分析,然后用非线性最小二乘拟合的方法,对所得各线性结果进行综合,以推求各种载荷作用(裂纹)下模态参数非线性变化规律.有关研究表明,将这种方法提供的结果用于桥梁承载力的动态法测定是合理可行的.     

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.     

Nonlinear Responses of Buckled Beams to Subharmonic-Resonance Excitations

We investigated theoretically and experimentally the nonlinear responseof a clamped-clamped buckled beam to a subharmonic resonance of orderone-half of its first vibration mode. We used a multi-mode Galerkindiscretization to reduce the governing nonlinear partial-differentialequation in space and time into a set of nonlinearly coupledordinary-differential equations in time only. We solved the discretizedequations using the method of multiple scales to obtain a second-orderapproximate solution, including the modulation equations governing itsamplitude and phase, the effective nonlinearity, and the effectiveforcing. To investigate the large-amplitude dynamics, we numericallyintegrated the discretized equations using a shooting method to computeperiodic orbits and used Floquet theory to investigate their stabilityand bifurcations. We obtained interesting dynamics, such as phase-lockedand quasiperiodic motions, resulting from a Hopf bifurcation,snapthrough motions, and a sequence of period-doubling bifurcationsleading to chaos. Some of these nonlinear phenomena, such as Hopfbifurcation, cannot be predicted using a single-mode Galerkindiscretization. We carried out an experiment and obtained results ingood qualitative agreement with the theoretical results.     

Experimental Nonlinear Identification of a Single Mode of a Transversely Excited Beam

A procedure is presented for using a primary resonance excitation in experimentally identifying the nonlinear parameters of a model approximating the response of a cantilevered beam by a single mode. The model accounts for cubic inertia and stiffness nonlinearities and quadratic damping. The method of multiple scales is used to determine the frequency-response function for the system. Experimental frequency- and amplitude-sweep data is compared with the prediction of the frequency-response function in a least-squares curve-fitting algorithm. The algorithm is improved by making use of experimentally known information about the location of the bifurcation points. The method is validated by using the extracted parameters to predict the force-response curves at other nearby frequencies.We then compare this technique with two other techniques that have been presented in the literature. In addition to the amplitude- and frequency-sweep technique presented, we apply a backbone curve- fitting technique and a time-domain technique to the second mode of a cantilevered beam. Differences in the parameter estimates are discussed. We conclude by discussing the limitations encountered for each technique. These include the inability to separate the nonlinear curvature and inertia effects and problems in estimating the coefficients of small terms with the time-domain technique.     

Identification of the state equation in complex non-linear systems

Building on the basic idea behind the Restoring Force Method for the non-parametric identification of non-linear systems, a general procedure is presented for the direct identification of the state equation of complex non-linear systems. No information about the system mass is required, and only the applied excitation(s) and resulting acceleration are needed to implement the procedure. Arbitrary non-linear phenomena spanning the range from polynomial non-linearities to the noisy Duffing-van der Pol oscillator (involving product-type non-linearities and multiple excitations) or hysteretic behavior such as the Bouc-Wen model can be handled without difficulty. In the case of polynomial-type non-linearities, the approach yields virtually exact results for sufficiently rich excitations. For other types of non-linearities, the approach yields the optimum (in least-squares sense) representation in non-parametric form of the dominant interaction forces induced by the motion of the system. Several examples involving synthetic data corresponding to a variety of highly non-linear phenomena are presented to demonstrate the utility as well as the range of validity of the proposed approach.     

Nonlinear vibration of beamplates with a delamination

The problem of beam-plates with a delamination located at an arbitrary site undergoing non-linear free vibration without damp is investigated. For this problem, nonlinear governing equations as well as the continuity, equilibrium and compatibility conditions are established. The Galerkin method and harmonic balance method are employed to find solutions. Finally, in some examples amplitude-frequency curves of the delaminated structure are presented in order to reveal some special dynamic features. Supported by the National Natural Science Foundation of China (No. 19872024).     

Normal Vibrations in Near-Conservative Self-Excited and Viscoelastic Nonlinear Systems

A perturbation methodology and power series are utilizedto the analysis of nonlinear normal vibration modes in broadclasses of finite-dimensional self-excited nonlinear systems closeto conservative systems taking into account similar nonlinear normal modes.The analytical construction is presented for some concretesystems. Namely, two linearly connected Van der Pol oscillatorswith nonlinear elastic characteristics and a simplesttwo-degrees-of-freedom nonlinear model of plate vibrations in agas flow are considered.Periodical quasinormal solutions of integro-differentialequations corresponding to viscoelastic mechanical systems areconstructed using a convergent iteration process. One assumesthat conservative systems appropriate for the dominant elasticinteractions admit similar nonlinear normal modes.     

On the Investigation of Some Parameter Identification and Experimental Modal Filtering Issues for Nonlinear Reduced Order Models

This paper discusses modal filtering of experimental data and the corresponding identification of linear and nonlinear parameters in reduced order space. Specifically, several experimental configurations will be discussed in order to provide insight into such identification issues as spatial discretization, observability, and the linear independence of the assumed filter or basis. The two experiments considered herein represent different measurement configurations of the same clamped–clamped beam. First, asymmetric inertial loading via asymmetric sensor location was considered, while the second scenario presents a symmetric sensor configuration. Several important conclusions can be drawn from the two experimental scenarios. First, by asymmetrically loading the beam, a corresponding asymmetric beam mode was excited yet not observable. In the second scenario, the symmetric distribution of sensors minimized the impact of the respective asymmetric mode. The resulting spatial information allowed for the proper filtering of the remnants of the asymmetric mode. Nonlinear parameters in modal space as well as the underlying linear parameters were successfully identified simultaneously in both experimental scenarios, although the usefulness of the asymmetrically loaded beam was limited. Finally, successful comparisons were made between the identified reduced order model and experimental response at the beam quarter point using the symmetric case and the beam midpoint using both experimental scenarios.     

自然弯扭梁的非线性分析

研究了两端边界均为完全约束的自然弯扭梁在小应变，大位移和大转动情况下的非线性性质，并考虑了横向剪切变形和扭转翘曲变形的影响，分析中还包括了拉伸，弯曲和扭转的各种弹性耦合。由最小势能原理可以导出所给问题的平衡方程。这里欧拉角可以用来表示任意大的转动。该方法还可方便地推广到其他各种不完全约束边界的情况。此外，利用上述结果还可以得到该梁在小位移理论中的基本方程和有关公式。     

轴压下各向异性开口薄壁杆的非线性共振

本文推导了轴压下各向异性薄壁直杆的非线性动力学方程组.在推导中计及了杆壁中面的剪切变形,以及开口薄壁杆中剪应力沿厚度变化而造成的Saint-Venant扭矩,但略去了杆壁的弯曲.从这一方程组出发,用渐近方法计算了简谐变化轴向压力作用下薄壁杆共振时幅频曲线,且对各种影响因素,包括Saint-Venant扭矩的影响,作了分析.     

机械振动信号盲源分离的时域方法

介绍盲源信号分离基本原理,分析了瞬态混叠模型和卷积混叠模型的特征,给出两个源信号独立性的评判准则,研究了盲分离正确性的评价误差函数,进行数值仿真,并用时域卷积瞬态混叠模型对实际机械振动信号进行源信号的盲分离。数值仿真和实际信号的分离效果较好,说明所用方法在机械振动信号盲源分离中的适用性。文中还分析了多源信号分离中所用方法的收敛性。