Eigenvalue-counting methods for non-proportionally damped systems

This paper presents numerical methods of counting the number of eigenvalues for non-proportionally damped system in some interested regions on the complex plane. Most of the eigenvalue analysis methods for proportionally damped systems use the well-known Sturm sequence property to check the missed eigenvalues when only a set of the lowest modes is used. However, in the case of the non-proportionally damped systems such as the soil–structure interaction system, the structural control system and composite structures, no counterpart of the Sturm sequence property for undamped systems has been established yet. In this study, a numerical method based on argument principle is explained with emphasis on the discretization of the contour and a new method based on Gleyse’s theorem is proposed. To verify the applicability of the methods, two numerical examples are considered.     

Fast stochastic analysis for non-proportionally damped system

Fundamental principles from structural dynamics, pseudo excitation method and pseudo force method are used to develop a new fast stochastic method for seismic response analysis of non-proportionally damped system. In the method, the mathematical equation of non-proportionally damped system is expressed in the iterative form, based on which the inverse operation of the matrices is avoided. Moreover, the new method does not need the solution of any complex eigenvalue problem, in contrast to other methods found in the literature.     

On damped linear dynamical systems

Summary This paper is concerned with damped linear dynamical systems which fail to satisfy the socalled convenience hypothesis, an assumption that guarantees the existence of classical normal modes. Under suitable conditions, estimates are given for the error committed when such a system is described approximately by a differential equation which does satisfy the convenience hypothesis. One of these estimates is used to derive an upper bound on the frequency response of systems which do not violate the convenience hypothesis too severely.

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Ein Beitrag zu den gedämpften, linearen, dynamischen Systemen

Übersicht Die Arbeit befaßt sich mit gedämpften, linearen, dynamischen Systemen, welche die Bequemlichkeitshypothese nicht erfüllen. Dabei wird untersucht, welchen Fehler man macht, wenn man eine allgemeine viskose Dämpfung durch eine zugeordnete modale Dämpfung entsprechend dieser Hypothese ersetzt. Eine der Fehlerabschätzungen wird verwendet zur Herleitung einer oberen Schranke der Antwortamplitude von Systemen, welche die Bequemlichkeitshypothese nicht allzu sehr verletzen.

     

非比例阻尼线性体系平稳随机地震响应计算的虚拟激励法

应用复振型分解方法,将非比例阻尼线性体系在地震作用下的动力方程求解问题转化为若干个广义复振子的求解与叠加问题。通过假定地震地面运动为一零均值的平稳随机激励,应用虚拟激励法原理,推导得到了广义复振子动力坐标的解析计算公式,进而得到了以复振型为基础的非比例阻尼线性体系随机地震响应计算的一般实数解析解答。算例证实了这种方法的可靠性及高效率。     

On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems

Recently Yuan and Agrawal [L. Yuan and O.P. Agrawal, A numerical scheme for dynamic systems containing fractional derivatives, Journal of Vibration and Acoustics 124 (2002) 321–324] presented a new numerical scheme to calculate the dynamic response of mechanical systems, the damping forces of which are described by fractional derivatives. When solving the resulting equation of motion by time integration, it is necessary to store the entire displacement history of the system due to the non-local character of the fractional derivatives. The cited scheme appears to overcome this drawback by transforming the equation of motion with the fractional term into a set of ordinary differential equations. It can be shown that this scheme is equivalent to a classical spring-dashpot representation and thus does not imply the benefits derived from fractional-derivative models. In addition, it is less flexible and incorrectly predicts the asymptotic behavior.     

On the methods of non-linear analysis of dynamical systems

This paper is concerned with the methods of non-linear analysis of dynamical systems and the associated bifurcation and stability problems. Attention is focused on the intrinsic harmonic balancing (IHB) technique, and the interrelationship between this technique and the methods of normal forms and averaging. Recent improvements and a complex formulation of the technique, which facilitates comparisons with other methods, are described. Thus, it is demonstrated that the simplified equations of an autonomous system, obtained by both the IHB and averaging techniques are identical, and these equations are, in fact, normal forms. Hilbert's 16th problem is analyzed as an illustrative example. It is observed that the IHB technique lends itself to a symbolic computer language (MAPLE) more efficiently compared to other methods; furthermore, its efficiency increases with the complexity of the system analyzed.     

Response analysis of fuzzy nonlinear dynamical systems

The transient and steady-state membership distribution functions (MDFs) of fuzzy response of a Duffing–Van der Pol oscillator with fuzzy uncertainty are studied by means of the fuzzy generalized cell mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established with a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. Fuzzy response is characterized by its topology in the state space and its possibility measure of MDFs. The evolutionary orientation of MDFs is in accordance with invariant manifolds toward invariant sets. In the evolutionary process of a steady-state fuzzy response with an increase of the intensity of fuzzy noise, a merging bifurcation is observed in a sudden change of MDFs from two sharp peaks of maximum possibility to one peak band around unstable manifolds.     

Modeling and analysis of a piezoelectric transducer embedded in a nonlinear damped dynamical system

This paper focuses on the dynamical analysis of the motion of a new three-degree-of-freedom (DOF) system consisting of two segments that are attached together. External harmonic forces energize this system. The equations of motion (EOM) are derived utilizing Lagrangian equations, and the approximate solutions up to the third order are investigated using the methodology of multiple scales. A comparison between these solutions and numerical ones is constructed to confirm the validity of the analytic solutions. The modulation equations (ME) are acquired from the investigation of the resonance cases and the solvability conditions. The bifurcation diagrams and spectrums of Lyapunov exponent are presented to reveal the different types of the system’s motion and to represent Poincaré maps. The piezoelectric transducer is connected to the dynamical system to convert the vibrational motion into electricity; it is one of the energy harvesting devices which have various applications in our practical life like environmental and structural monitoring, medical remote sensing, military applications, and aerospace. The influences of excitation amplitude, natural frequency, coupling coefficient, damping coefficient, capacitance, and load resistance on the output voltage and power are performed graphically. The steady-state solutions and stability analysis are discussed through the resonance curves.

     

Wave analysis and control of double cascade-connected damped mass-spring systems

This paper presents wave analysis and control for double cascade-connected damped mass-spring systems, whose mass is connected beyond the adjacent masses. The system is motivated by a cantilevered tensegrity beam supporting tensile and compressive forces. The wave solution is derived from a recurrent formula, and the properties of the propagation constants are precisely investigated. Elimination of reflected waves provides the impedance matching controller. We show that the impedance matching controller can be constructed from a similarity transformation of the characteristic impedance matrix by a matrix composed of the propagation constants. A numerical example of vibration control of a tensegrity beam is shown.     

基于微分求积法求解非粘滞阻尼系统的时程响应

研究一种含有指数型非粘滞阻尼线性多自由度振动系统的时程分析问题。该非粘滞阻尼模型假设阻尼力与质点速度的时间历程相关,数学表述为质点速度与核函数的卷积。由于阻尼模型的改变,常用的数值积分方法（如Newmark-β法、Wilson-θ法）不能直接应用于这种非粘滞阻尼系统。基于一种无条件稳定的微分求积方法,给出了这种非粘滞阻...     

非线性随机动力系统的概率密度演化分析

阐述了基于概率密度演化理论进行多自由度结构非线性随机动力反应分析的基本思想.采用随机过程的正交分解或物理系统建模的思想,实现随机激励的随机函数表述.对由此获得的随机状态方程采用概率密度演化理论求解,可以获得随机动力系统反应的概率密度函数及其演化.以某剪切型框架结构的非线性随机地震反应分析为例,说明了所发展方法的可行性和有效性.     

具有奇异位置的多体系统动力学方程的隐式算法

本文研究了在运动过程中具有奇异位置的多体系统动力学方程的隐式算法，给出了隐式算法所用的Ｊａｃｏｂｉ矩阵，并建立了该矩阵中各子矩阵间的计算关系，提高了计算效率，计算结果表明隐式算法的计算速度和精度明显优于显式算法。     

Asymptotic analysis of kinematically excited dynamical systems near resonances

The dynamic response of a harmonically and kinematically excited spring pendulum is studied. This system is a multi-degree-of-freedom system and is considered as a good example for several engineering applications. The multiple-scale (MS) method allows us to analytically solve the equations of motion and recognize resonances. Also stability of the steady-state solutions can be verified. The transfer of energy from one to another mode of vibrations is illustrated.     

Bounds on the forced response of damped linear systems

Bounds are obtained on the magnitude of the displacement vector representing the response of a damped linear mechanical system to an oscillatory forcing function. Several restrictions used in previous treatments are avoided, particularly to the effect that the system modes must satisfy a decoupling condition.     

Periodic solutions of dynamical systems

Summary In this paper we look for T-periodic solutions of dynamical systems. Particularly we consider the system whereU ɛC ¹(ℝ ⁿ x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ ⁿ , ∀t ɛ ℝ T>0. We assume that the problem is asymptotically linear with a bounded nonlinearity. Under a resonance assumption, we find a multiplicity of T-periodic solutions for T large enough.

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Sommario In questo lavoro si cercano soluzioni periodiche di periodo T assegnato di sistemi dinamici. In particolare si considera un sistema di n equazioni differenziali del secondo ordine del tipo doveU ɛC ¹(ℝ ⁿ x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ ⁿ , ∀t ɛ ℝ T>0. Nel caso in cui il problema sia asintoticamente lineare, con termine nonlineare limitato e in condizioni di risonanza, troviamo che esiste tale che per il sistema ha una molteplicità di soluzioni.

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Presented at the VII A.I.M.E.T.A. and supported by M.P.I. (40% and 60%).