Methods of Identification of Nonlinear Mechanical Vibrating Systems

Methods for determination of the dynamic characteristics and parameters of mechanical vibrating systems by processing experimental data on controlled vibrations are presented. These methods are intended for construction of mathematical models of objects to be identified and classed as parametric and nonparametric methods. The quadrature formulas of the nonparametric-identification method are derived by inverting the integral parameters of approximate analytical solutions of nonlinear differential equations. The parametric-identification method involves setting up and solving systems of linear algebraic equations in the sought-for inertia, stiffness, and dissipation parameters by integrating experimental processes using special weighting functions. Depending on the type of the nonlinearity of the vibrating system and the method of representing experimental processes, the weighting functions can be oriented toward displacement, velocity, or acceleration gauges. The results of studies made mainly at the Institute of Mechanics of the National Academy of Sciences of Ukraine are presented     

The Dynamics of Discrete Mechanical Systems with Perfect Unilateral Constraints

The dynamics of discrete mechanical systems with perfect unilateral constraints is formulated in a very general setting. The well-posedness of the resulting evolution problem is studied. It is proved that existence and uniqueness of a maximal solution is ensured provided strong assumptions are made on the regularity of the data: they are supposed to be analytic. Simple examples show that this regularity assumption may not be relaxed. Sufficient conditions to ensure that the maximal solution is defined for all time are supplied. The continuous dependence of the solution on initial conditions is also studied and the numerical computation of the solution is discussed. Accepted April 18, 2000?Published online September 18, 2000     

Modeling and Algorithm on a Class of Mechanical Systems with Unilateral Constraints

In the present paper, the cable-structures are considered as a class of mechanical complementary-slackness systems. Based on the optimization algorithms used for multi-body dynamics with unilateral contacts, an algorithm by means of artificial neural network (NNW) is developed. The following two classes of cable-structures have been considered force-elongation of cable member follows elastic behavior and work-hardening assumption. Due to simplicity the former is used to prove the method reliability, and the latter, as general cable-structure problem is handled. First, the complementarity problems for those structures have been formulated; then using generalized Gaussian‘ least action principle they are summarized as an optimization problem. Based on Hopfield’s work, an artificial NNW has been designed and used to decide combination of possible constraints at each step in simulation. As examples, two cable-structures have been investigated. The calculated results for a simple suspension structure evidence the reliability and time-economization of the proposed method. An example of guyed mast shows the suitability of the proposed method for practical cable-structures.This paper supported by NSFC, No.10072040     

Sweep Tests on Vibrating Systems with Power-Law Damping

The non-stationary response in vibrating systems with power-law damping, subjected to sweep tests is considered. Using the method of averaging, it is shown that the bifurcation delay persists in all forms of power-law damping, viz., Coulomb, orifice and cubic damping models. Experimental investigations on a soft type of isolator confirm the existence of delay in the jump phenomenon. These results also indicate that the softening nature of non-linearity can be profitably employed in the design of isolators and vehicle suspensions.     

Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints

This paper is concerned with the dynamics of a mechanical system subject to nonintegrable constraints. In the first part, we prove the equivalence between the classical nonholonomic equations and those derived from the nonholonomic variational formulation, proposed by Kozlov in [10–12], for a class of constrained systems with constraints transverse to a foliation. This result extends the equivalence between the two formulations, proved for holonomic constraints, to a class of linear nonintegrable ones. In the second part, we derive the nonholonomic variational reduced equations for a constrained system with symmetry and constraint transverse to a principal bundle fibration, using a reduction procedure similar to the one developed in [5]. The resulting equations are compared with the nonholonomic reduced ones through mechanical examples.     

Averaging of Noisy Nonlinear Systems with Rapidly Oscillating and Decaying Components

We consider a noisy n-dimensional nonlinear dynamical system containing rapidly oscillating and decaying components. We extend the results of Papanicolaou and Kohler and Namachchivaya and Lin; these results state that as the noise becomes smaller, a lower dimensional Markov process characterizes the limiting behavior. Our approach springs from a direct consideration of the martingale problem and considers both quadratic and cubic nonlinearities.     

带约束非线性多体系统动力学方程数值分析方法

Lagrange方法是建立带约束多体系统动力学方程的普遍方法之一 ,其方程的形式为微分 代数方程组 ,数值计算与数值分析是研究多体系统动力学特性的重要方法。本文利用缩并法给出了带约束多体系统动力学方程的隐式数值计算方法和Lyapunov指数的计算方法。将数值仿真、Lya punov指数计算和Poincare映射有机结合 ,分析非线性多体系统动力学行为。通过一个算例 ,说明该方法的有效性     

Mechanical Fatigue Measurement via a Vibrating Cantilever Beam for Self-Supported Thin Solid Films

In this paper, we develop a novel experimental apparatus, referred to as the resonant frequency device, and establish methodology to measure the fatigue properties of thin solid films. Arranging thin-film strips of our specimens into the mechanical setting of a cantilever beam and using state-of-the-art piezo actuators to generate oscillation at the clamp of the cantilever, we create a system suitable for studying the material properties of the cantilever, such as Young's modulus, fatigue and possibly, loss tangent. Deformation of the cantilever is our controlled variable in the present study, and measured with fiber-optic probes pointed at the specimen and at the piezo driver. Stress is calculated from relative deformation of the cantilever specimen with respect to the piezo actuator via a photograph of the cantilever under vibration with a curve fitting method. A LabView computer program is developed for the fatigue tests to accurately count number of cycles applied on the specimens, and a feedback mechanism is adopted to maintain displacement during the tests. Here, we present our experimental setup, procedure and theoretical models for material-property extraction. For small displacement, the two-dimensional Euler–Bernoulli beam theory is adopted. With large displacement, the system behaves as the Duffing oscillator due to geometrical nonlinearity. In addition, some experimental observations of the piezo actuators and fiber optics are reported. The method is applied to evaluate the fatigue properties of nanolayered copper-niobium composites and significant increase in the fatigue endurance limit compared to the constituent materials in the bulk form is noted.     

Vibration Problems in Selected Torsional Mechanical Systems

The paper deals with vibrations of mechanical systems torsionally deformed. These problems can be discussed using two-dimensional or one-dimensional models. After the presentation of basic equations for two-dimensional problems, the study is focused on one-dimensional problems for discrete-continuous systems with a local nonlinearity and on the effect of the local nonlinearity on the behaviour of these systems.     

Determination of the Chaos Onset in Mechanical Systems with Several Equilibrium Positions

Determination of the chaos onset in some mechanical systems with several equilibrium positions are analyzed. Namely, the snap-through truss and the oscillator with a nonlinear dissipation force, under the external periodical excitation, are considered. Two approaches are used for the chaos onset determination. First, Padé and quasi-Padé approximants are used to construct closed homoclinic trajectories for a case of small dissipation. Convergence condition used earlier in the theory of nonlinear normal vibration modes as well conditions at infinity make possible to evaluate initial amplitude values for the trajectories with admissible precision. Mutual instability of phase trajectories is used as criterion of chaotic behavior in nonlinear systems for a case of not small dissipation. The numerical realization of the Lyapunov stability definition gives us a possibility to observe a process of appearance and fast enlargement of the chaotic behavior regions if some selected parameters of the dynamical systems under consideration are changing.     

Nonlinear Dynamics of Mechanical Systems Under Complicated Forcing

This paper considers the problem of simple linear mechanical oscillators forced by complicated forcing functions, such as those induced by fluids or humans. Such problems arise in civil engineering in the design of bridges, grandstands and towers. The paper questions whether low dimensional models have much to contribute to the understanding of such problems, given the complicated possibilities of high-dimensional forcing.     

柔性对多体系统中铰接副磨损的影响

通过集成柔性多体动力学与磨损计算程序,提出了一种用于对柔性多体系统中间隙铰接副部位的磨损进行了预测的方法.基于绝对节点坐标方法(ANCF)建立了柔性部件的多体动力学模型,引入Lankanrani和Nikravesh提出的连续接触力模型计算间隙铰接副部分的法向接触力,采用Lu Gre摩擦模型计算切向摩擦力,并利用基于Archard模型的迭代计算程序计算磨损.为了提高计算效率,引入了并行计算策略.最后,通过对一个含柔性连杆的曲柄滑块机构机构进行仿真计算,发现当考虑部件的柔性时,得到的间隙处的冲击力会大幅降低,预测的磨损量也随之降低,并且随着机构柔性的增强,这种效果更为明显.     

Vibrating triangular plate

Summary The problem solved is that of determining the small transverse vibrations of a thin elastic plate in the shape of a 30° right triangle. The plate is simply supported on all sides, and started with a prescribed initial velocity.     

变质量完整力学系统的Hojman守恒量

研究完整力学系统由Noether对称性导致的Hojman守恒量．列写系统的运动微分方程；在时间不变的特殊无限小变换下，研究系统的Noether对称性与Lie对称性，给出Noether对称性为Lie对称性的条件；将Hojman定理推广至变质量系统，并举例说明结果的应用．     

完整力学系统的共形不变性与守恒量

将Birkhoff方程的共形不变性和共形因子的概念拓展到完整力学系统,研究一般完整力学系统在无限小变换下的共形不变性与守恒量.给出了一般完整力学系统的共形不变性的定义和确定方程;研究了系统的Noether对称性与共形不变性之间的关系,研究表明,当Noether对称变换的生成元和非势广义力满足一定条件时,变换也是共形不变的,给出了相应的共形因子表达式,得到了一般完整力学系统的共形不变性直接导致的Noether守恒量;研究了系统的Lie对称性与共形不变性之间的关系,给出了与Lie对称性相应的无限小变换共形不变的充分必要条件,得到了一般完整力学系统的共形不变性直接导致的Lutzky守恒量.文中还举例说明结果的应用.     

Delay Equations with Rapidly Oscillating Stable Periodic Solutions

We prove analytically that there exist delay equations admitting rapidly oscillating stable periodic solutions. Previous results were obtained with the aid of computers, only for particular feedback functions. Our proofs work for stiff equations with several classes of feedback functions. Moreover, we prove that for negative feedback there exists a class of feedback functions such that the larger the stiffness parameter is, the more stable rapidly oscillating periodic solutions there are. There are stable periodic solutions with arbitrarily many zeros per unit time interval if the stiffness parameter is chosen sufficiently large.     

受控振动系统时滞依赖稳定性判据的LMI方法

针对带有不确定性时滞的振动控制系统,引入改进的二向量内积的上确界,对Lyapunov导数进行合理放大,提出了一种新的时滞依赖鲁棒性稳定准则。根据这种新的判别方法,受控振动系统的最大时滞量可以通过Matlab控制工具箱LMI(线性矩阵不等式)得到。讨论了单自由度系统中最大时滞随系统参数的变化情况及多自由度系统中控制器最优权值R的确定。算例结果表明,在允许时滞范围内,经典的LQR控制器可以收到令人满意的效果。