On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping

We study the asymptotic stability of a dissipative evolution in a Hilbert space subject to intermittent damping. We observe that, even if the intermittence satisfies a persistent excitation condition, if the Hilbert space is infinite-dimensional then the system needs not being asymptotically stable (not even in the weak sense). Exponential stability is recovered under a generalized observability inequality, allowing for time-domains that are not intervals. Weak asymptotic stability is obtained under a similarly generalized unique continuation principle. Finally, strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition, evaluating their total contribution to the decay of the trajectories of the damped system. Our results are discussed using the example of the wave equation, Schrödinger?s equation and, for strong stability, also the special case of finite-dimensional systems.     

On the Asymptotic Stability of Wave Equations Coupled by Velocities of Anti-symmetric Type

In this paper, the authors study the asymptotic stability of two wave equations coupled by velocities of anti-symmetric type via only one damping. They adopt the frequency domain method to prove that the system with smooth initial data is logarithmically stable, provided that the coupling domain and the damping domain intersect each other.Moreover, they show, by an example, that this geometric assumption of the intersection is necessary for 1-D case.     

Resonant vibrations in harmonically excited weakly coupled mechanical systems with cyclic symmetry

The resonant vibrations in weakly coupled nonlinear cyclic symmetric structures are studied. These structures consist of weakly coupled identical nonlinear oscillators. A careful bifurcation analysis of the amplitude equations is performed in the fundamental resonance case for an illustrative example consisting of a three particle system. In case of a uniformly distributed excitation, a localized response is identified in which one of the particles exhibits large amplitude motions compared to those of the other particles. In case of single-particle excitation, it is found that for very small coupling strength and large external mistuning, a large stable localized periodic response coexists with an extended small response. With an increase in the coupling strength, multiple extended solutions arise near the exact external resonance via saddle-node bifurcations. Further increase in coupling strength and a decrease in damping results in isolated asymmetric solution branches, which bifurcate from the symmetric solutions via symmetry-breaking bifurcations. The role of coupling strength in creating/destroying localized solutions is discussed.     

Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beams with indefinite damping

We study damped Euler–Bernoulli beams that have nonuniformthickness or density. These nonuniformfeatures result in variablecoefficient beam equations. We prove that despite the nonuniformfeatures, the eigenfunctions of the beam form a Riesz basisand asymptotic behaviour of the beam system can be deduced withoutany restrictions on the sign of the damping. We also providean answer to the frequently asked question on damping: ‘howmuch more positive than negative should the damping be withoutdisrupting the exponential stability?’, and result ina criterion condition which ensures that the system is exponentiallystable.     

Non-uniform asymptotic stability for the damped linear oscillator

Sufficient conditions are established for non-uniform asymptotic stability of a linear oscillator with damping term. The obtained results clarify a difference between the uniform asymptotic stability and the asymptotic stability. Some simple examples are included to illustrate the results. Especially, Bessel’s differential equations are taken up and it is proved that the equilibrium is asymptotically stable, but it is not uniformly asymptotically stable.     

Resonance dynamics of nonlinear flutter systems

We consider a special class of nonlinear systems of ordinary differential equations, namely, the so-called flutter systems, which arise in Galerkin approximations of certain boundary value problems of nonlinear aeroelasticity and in a number of radiophysical applications. Under the assumption of small damping coefficient, we study the attractors of a flutter system that arise in a small neighborhood of the zero equilibrium state as a result of interaction between the 1: 1 and 1: 2 resonances. We find that, first, these attractorsmay be both regular and chaotic (in the latter case, we naturally deal with numerical results); and second, for certain parameter values, they coexist with the stable zero solution; i.e., the phenomenon of hard excitation of self-oscillations is observed.     

Controlling the nonlinear dynamics of a beam system

Control based on linear error feedback is applied to reduce vibration amplitudes in a piecewise linear beam system. Hereto small amplitude 1-periodic solutions are stabilized wherever they coexist with two or more long-term solutions. In theory, no control effort is required to maintain the 1-periodic response once it has been stabilized. For the beam system, 1-periodic solutions are stabilized by feedback at one location along the beam. Feedback is represented by servo-stiffness or servo-damping which results from increasing two corresponding control parameters. At appropriate levels of these parameters local, or global, asymptotic stability (of the zero-equilibrium) of the error dynamics, i.e. stability of the underlying 1-periodic solutions, can be guaranteed. Local asymptotic stability can be guaranteed for a large range of actuator locations and excitation frequencies and is indicated by bifurcations. Global asymptotic stability can only be guaranteed for a limited range of actuator locations on the basis of the well-known circle criterion. The difference between local and global asymptotic stability in terms of the required values for the control parameters can be significant, and may result in large differences in control performance.     

Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation

The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order α (0 < α < 1) subject to the narrow-band random parametric excitation are investigated. The method of multiple scales is developed to derive two first order stochastic differential equation of amplitude and phase, and then to examine the influences of fractional order and intensity of random excitation on the first-order and second-order moment. As an example, the stochastic Duffing oscillator with fractional derivative damping is considered. The effects of detuning frequency parameter, the intensity of random excitation and the fractional order derivative damping on stability are studied through the largest Lyapunov exponent. The corresponding theoretical results are well verified through direct numerical simulations. In addition, the phenomenon of stochastic jump is analyzed for parametric principal resonance responses via finite differential method. The stochastic jump phenomena indicates that the most probable motion is around the larger non-trivial branch of the amplitude response when the intensity of excitation is very small, and the probable motion of amplitude responses will move from the larger non-trivial branch to trivial branch with the increasing of the intensity of excitation. Such stochastic jump can be considered as bifurcation.     

集中阻尼弦本征解的性质

利用Dirac δ函数,在全域建立并求解集中阻尼弦的动力学方程,导出其本征方程组、频率方程和本征函数的一般形式,推导了单项阻尼下本征函数的具体形式,并分析了中点阻尼对本征解的影响.同时,讨论了混合动力学系统在频率 阻尼关系、衰减率和完全抑制振动的最优阻尼3个方面既不同于连续系统,又不同于离散系统的特性:1）系统频率与其阻尼无关;2）各阶本征函数在单位时间内的衰减率都相同,衰减率与本征值的阶次无关;3）当阻尼取2时,系统衰减率趋于无穷大,系统不能发生任何有阻尼振动.     

Stabilizing the pull-in instability of an electro-statically actuated micro-beam using piezoelectric actuation

In the present article an investigation is presented into the stability of an electro-statically deflected clamped–clamped micro-beam sandwiched by two piezoelectric layers undergoing a parametric excitation applying an AC voltage to these layers. Applying an electrostatic actuation not only deflects the micro-beam but also decreases the bending stiffness of the structure, which can lead the structure to an unstable position by undergoing a saddle node bifurcation. Utilizing an appropriate AC actuation voltage to the piezoelectric layers produces a time varying axial force, which can play the role of a stabilizer exciting the system parameter. The governing equation of the motion is a nonlinear electro-mechanically coupled type PDE, which is derived using variational principle and discretized, applying Eigen-function expansion method. The resultant is a Mathieu type equation in its damped form. Using Floquet theory for single degree of freedom system the stable and unstable regions of the problem are investigated. The effects of viscous damping and electrostatic actuation on the stable regions of the problem are also studied.     

Asymptotic analysis of steady solutions of the KdVB equation with application to resonant sloshing

The forced Korteweg-de Vries equation with Burgers’ damping (fKdVB) on a periodic domain, which arises as a model for water waves in a shallow tank with forcing near resonance, is considered. A method for construction of asymptotic solutions is presented, valid in cases where dispersion and damping are small. Through variation of a detuning parameter, families of resonant solutions are obtained providing detailed insight into the resonant response character of the system and allowing for direct comparison with the experimental results of Chester and Bones (1968).     

Exponential Decay Rate for a Timoshenko Beam with Boundary Damping

The exponential decay rate of a Timoshenko beam system with boundary damping is studied. By asymptotically analyzing the characteristic determinant of the system, we prove that the Timoshenko beam system is a Riesz system; hence, its decay rate is determined via its spectrum. As a consequence, by showing that the imaginary axis neither has an eigenvalue on it nor is an asymptote of the spectrum, we conclude that the system is exponentially stable.     

Algebraic Approach to the Stabilization of a Differential System of Retarded Type

For a spectrally controllable linear autonomous systems with commensurable delays, we construct state feedbacks ensuring the complete damping of the original system (finite stabilization) as well as the complete damping of the original system and the asymptotic stability of the closed-loop system (complete stabilization). The spectral reduction and asymptotic stabilization problems are considered as auxiliary problems. The argument is constructive, and the results are illustrated by an example.     

Stabilization of the stationary motions of non-holonomic mechanical systems

The possible stabilization of the unstable stationary motions of a non-holonomic system is studied from the standpoint of general control theory /1, 2/. As distinct from the case previously considered /3/, when forces of a certain structure are applied with respect to both positional and cyclical coordinates, the stabilization is obtained here by applying control forces only with respect to the cyclical coordinates /4/; the control forces may be applied with respect to some or all of the cyclical coordinates, and depend on the positional coordinates, the velocities, and the corresponding cyclical momenta. It is shown that, just as in the case of holonomic systems /5, 6/, depending on the control properties of the corresponding linear subsystem, the stationary motions, whether stable or unstable, can be stabilized, up to asymptotic stability with respect to all the phase variables, or asymptotic stability with respect to some of the phase variables and stability with respect to the remaining variables. The type of stabilization with respect to the given phase variables depends on the Lyapunov transformations which are needed in order to reduce the critical cases obtained to singular cases /7, 8/.     

第五类Painlevé方程解的渐近性态分析

本文用比较直接的方法研究Painleve方程的渐近解和连同公式:(1)先求出数值解,然后用最小二乘法拟合出最佳渐近解;(2)根据最佳渐近解的表达形式,用谐波平衡法得到振荡渐近解与参数之间的依赖关系,即连同公式．当参数α,β,γ和δ满足一些条件时,对一般实的第五类Painleve方程,我们找出了振荡渐近解和连同公式．     

Sharp Estimates for the Number of Degrees of Freedom for the Damped-Driven 2-D Navier-Stokes Equations

We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier-Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier-Stokes system with damping, our estimates for the number of the determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. Our investigation of the damped-driven 2-D Navier-Stokes system is inspired by the Stommel-Charney barotropic model of ocean circulation where the damping represents the Rayleigh friction. We remark that our results equally apply to the viscous Stommel-Charney model.     

Flexible-link robots with combined trajectory tracking and vibration control

This work deals with asymptotic trajectory tracking and active damping injection on a flexible-link robot by application of Multiple Positive Position Feedback. The flexible-link robot is modeled and validated by using finite element methods and experimental modal analysis, and then a reduced order model of the flexible-link robot dynamics, up to the first dominant vibration modes, is employed for experimental evaluation on a test rig. Then, a combined control scheme is synthesized in two parts: first, a Sliding-Mode Control based on a cascaded Proportional-Integral-Derivative for regulation and trajectory tracking tasks, via a direct current motor torque as the control input for the overall system dynamics, and, second, a Multiple Positive Position Feedback for active vibration control and attenuation of residual vibrations on the tip position, via the input voltage applied to a piezoelectric patch actuator attached directly on the flexible beam. The results are evaluated on an experimental platform, where the dynamic performance of the overall active vibration control scheme leads to fast and effective tracking results, with damping ratios increased up to 300%.     

Dynamics of A Damping Oscillator with Impact and Impulsive Excitation

There exist many types of external excitations that make the damping oscillator with impact have complex dynamics. In this study, both external impulsive excitation and impact are considered to construct a vibro-impact system. The fixed time pulse (impulsive excitation) and the state pulse (impact) lead to the complex and interesting dynamics. The conditions of the existence and stability of four kinds of periodic solutions are investigated, and the bifurcations of period-(1, 0) and period-(1, 1) solutions are analytically studied. Numerical simulations on periodic solutions and bifurcation diagrams are shown by the illustrative example.     

非传播孤立波及其相互作用的数值模拟

通过数值求解由Miles导出的目前公认的的非传播孤立波的控制方程——一个带复共轭项的非线性立方SchrLdinger方程,对非传播孤立波进行研究。讨论了Miles方程中的线性阻尼系数α的值,计算表明,线性阻尼α对形成稳定的非传播孤立波影响很大,Laedke等人关于非传播孤立波的稳定性条件只是一个必要条件,而不是充分条件。模拟了两个非传播孤立波的相互作用,数值模拟表明,两个波的作用模式依赖于系统的参数,对不同的初始扰动及其演化的计算表明,只有适当的初始扰动才能形成单个稳定的非传播孤立波,否则扰动可能消失或发展成多个孤立波。     

Forced nonlinear oscillation of underactuated systems: the balance board example

In this contribution we consider a mechanical model of the so-called balance board or “indoboard” system, which is an unstable underactuated mechanical system. Instead of stabilizing one equilibrium point we investigate the asymptotic stabilization of a closed orbit, which results in a nonlinear oscillation. To this end the system is considered in coordinates of the real Jordan normal form, which facilitates the construction of a Control-Lyapunov function. On this basis, a nonlinear feedback law which renders the periodic orbit asymptotically stable is easily derived via Sontags formula. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)