BFEEDBACK CONTROL OF SINGLE-LINK FLEXIBLE ARMS

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Exponential stability of a kind of wave equation with boundary feedback control

The problem of exponential stability of a kind of wave equation with damping and boundary output feedback control is investigated. The spectral structure of the system operator is analyzed and it is shown that the c0-semigroup generated by the system operator is exponential stable if only the coefficients viscous damping and boundary feedback control are not zeros simultaneously.     

Exponential Decay Rates for Structural Acoustic Model with an Overdamping on the Interface and Boundary Layer Dissipation

We study uniform stability properties of a strongly coupled system of Partial Differential Equations of hyperbolic/parabolic type, which arises from the analysis and control of acoustic models with structural damping on an interface. A challenging feature of the present model is the presence of additional strong boundary damping which is responsible for lack of uniform stability of the free system ( overdamping phenomenon). It has been shown recently that by applying full viscous damping in the interior of the domain and suitable static damping on the interface, then the corresponding feedback system is uniformly stable. In this article we prove that uniform decay rates of solutions to the system can be achieved even if viscous damping is active just in an arbitrary thin layer near the interface.     

Single-mode control and chaos of cantilever beam under primary and principal parametric excitations

A non-linear control law is proposed to suppress the vibrations of the first mode of a cantilever beam when subjected to primary and principal parametric excitations. The dynamics of the beam are modeled with a second-order non-linear ordinary-differential equation. The model accounts for viscous damping air drag, and inertia and geometric non-linearities. A control law based on quantic velocity feedback is proposed. The method of multiple scales method is used to derive two-first ordinary differential equations that govern the evolution of the amplitude and phase of the response. These equations are used to determine the steady state responses and their stability. Amplitude and phase modulation equations as well as external force–response and frequency–response curves are obtained. Numerical simulations confirm this scenario and detect chaos and unbounded motions in the instability regions of the periodic solutions.     

The stability of an inverted pendulum with a vibrating suspension point

The problem of stabilizing the upper vertical (inverted) position of a pendulum using vibration of the suspension point is considered. The periodic function describing the vibrations of the suspension point is assumed to be arbitrary but possessing small amplitudes, and slight viscous damping is taken into account. A formula is obtained for the limit of the region of stability of the solutions of Hill's equation with damping in the neighbourhood of the zeroth natural frequency. The analytical and numerical results are compared and show good agreement. An asymptotic formula is derived for the critical stabilization frequency of the upper vertical position of the pendulum. It is shown that the effect of viscous damping on the critical frequency is of the third-order of smallness and, in all the examples considered, when viscous damping is taken into account the critical frequency increases.     

The stability of the rotation of a heavy body with a viscous filling

The stability of the permanent rotation of a symmetrical heavy body with a viscous filling is investigated. A finite-dimensional phenomenological model of the “internal friction” with which the filling acts on the wall of the cavity is constructed based on the Helmholtz equations for a vortex. The boundaries of the stability limit are constructed and the interaction between the internal friction and the external damping is tracked. It is shown that the cases of a cavity that is oblate and prolate along the axis of rotation lead to the existence of different forms of stability regions.     

Stabilization of the dynamic elastica only by damping torque

We consider a system of partial differential equations called the dynamic elastica, which describes the motion of a geometrically nonlinear (i.e., largely deflecting) elastic beam and is derived based on Euler–Bernoulli’s assumption. The aim of the paper is to examine the effect of damping torque (i.e., external torque generated as a negative feedback of the angular velocity of the beam’s centerline) on the stability of the elastica.     

The limiting steady rotations of a body on a string with a suspension point on the axis of symmetry

The steady motions of an axisymmetrical rigid body suspended from a fixed base by a weightless undeformable rod or a non-twisting inextensible string are investigated. The case when the rod is fastened to the body at a point situated on its axis of dynamic symmetry is considered. All types of limiting equilibrium configurations which are possible when there is an unlimited increase in the angular velocity of rotation of the system about the vertical are analysed. Domains in which each type of limiting regular precession and permanent rotation can exist are constructed in the space of dimensionless parameters, and the nature of their asymptotic behaviour when the angular velocity increases is determined. The limiting motions which are possible in the case of suspension on a rod and impossible in the case of suspension on a string are investigated.     

The vibration of a rotating heavy non-uniform thread and its stability

The linear vibration of a heavy non-uniform thread is investigated for different boundary conditions at the ends and taking an arbitrary additional tension into account. The thread is assumed to be ideal and inextensible and the motion takes place in a plane which may rotate about the vertical axis at constant angular velocity. A general scheme for solving the initial- and boundary value problem is proposed. Attention is focused mainly on the effective computation of the natural frequencies and mode of vibration. Given specific parametric types of mass distribution for the thread, sufficiently complete families of solutions describing the principal modes of vibration are constructed. Based on these families, stability and instability domains are constructed effectively, in terms of the system parameters, for a plane vibration of a rotating heavy thread subject to concentrated tension. New mechanical effects, of possible interest in practice, are observed and discussed.     

Bogdanov–Takens bifurcation in an oscillator with negative damping and delayed position feedback

In this paper, Bogdanov–Takens bifurcation occurring in an oscillator with negative damping and delayed position feedback is investigated. By using center manifold reduction and normal form theory, dynamical classification near Bogdanov–Takens point can be completely figured out in terms of the second and third derivatives of delayed feedback term evaluated at the zero equilibrium. The obtained normal form and numerical simulations show that multistability, heteroclinic orbits, stable double homoclinic orbits, large amplitude periodic oscillation, and subcritical Hopf bifurcation occur in an oscillator with negative damping and delayed position feedback. The results indicate that negative damping and delayed position feedback can make the system produce more complicated dynamics.     

Comparison of Numerical Simulation Models for Open Loop Flight Simulations in the Human Centrifuge

To achieve predefined Gz load profile in high G training in human centrifuge, it is necessary to determine angular velocity and acceleration of a planetary axis (centrifuge arm). Initial value problem that can not be solved in closed form was obtained. Several discretization methods for calculating angular velocity of centrifuge arm driven by DC servo motor are presented. Simulations are performed for different positive and negative values of Gz onset. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global stability of a Leslie–Gower predator–prey model with feedback controls

In this work the global stability of a unique interior equilibrium for a Leslie–Gower predator–prey model with feedback controls is investigated. The main result together with its numerical simulations shows that feedback control variables have no influence on the global stability of the Leslie–Gower model, which means that feedback control variables only change the position of the unique interior equilibrium and retain its global stability.     

Stability of digital force control in the presence of damping

In the plain of the proportional gain and the ratio of the natural and sampling frequencies, the stability charts of digital force control are constructed for one degree of freedom mechanical models with large variety of viscous damping parameters. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非线性梁结构的参数振动稳定性及其主动控制

采用压电材料研究了参数激励非线性梁结构的运动稳定性及其主动控制,通过速度反馈控制算法获得主动阻尼,利用Hamilton原理建立含阻尼的立方非线性运动方程,采用多尺度方法求解运动方程获得稳定性区域．通过数值算例,分析了控制增益、外激振力幅值等因素对稳定性区域和幅频曲线特性的影响．分析表明:控制增益增大,结构所能承受的轴向力也增大,在一定范围内结构的主动阻尼比也增加;随着控制增益的增大,响应幅值逐渐降低,但所需的控制电压存在峰值点．     

Stabilization of a nonlinear flow-plate interaction via component-wise decomposition

Asymptotic-in-time interior feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [8] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of energy dissipation the plate dynamics converge to a compact and finite dimensional set [6, 7]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the set of stationary states in the natural energy topology. We show a decomposition of the dynamics into “smooth” component and global-in-timeHadamard continuous component, thus permitting approximation by smooth data. That the flows are subsonic is critical for our approach. Our result implies that flutter (a periodic or chaotic end behavior) is not present in subsonic flows with sufficient viscous damping in the structure.     

Stabilization of Euler- Bernoulli Beam by A Boundary Control

A flexible beam is considered. One end of the beam is free and another end is clamped under changeable angle, with angular velocity taken as control variable. A suitable stability criterion is introduced and the uniform exponential stabilizability is established for a linear feedback control applied at the clamped end.     

Mechanical model of a mistuned 2DOF-structure coupled by viscous damping

In this paper the vibration amplitude reduction of a mechanical system is investigated when applying a damper between different DOF of that structure. Thereby, an elementary mechanical model consisting of two mass-spring elements with slightly differing eigenfrequencies and which are connected via a viscous dashpot damper is regarded. To investigate the systems behavior, an analysis of the transfer function and its poles and zeros in the L APLACE domain is performed when varying the damping constant. This reveals that the achievable amplitude reduction depends on the mistuning of the system and that even ideal viscous damping leads to a coupling which results in a shift of eigenfrequencies. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stokes flow between eccentric rotating spheres with slip regime

The steady axisymmetric flow problem of a viscous fluid contained between two eccentric spheres that rotate about an axis joining their centers with different angular velocities is considered. A linear slip of Basset-type boundary condition at both surfaces of the spherical particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in the spherical coordinate systems based on the inner solid particle and the spherical container. The boundary conditions on the particle’s surface and spherical container are satisfied by a collocation technique. Numerical results for the coupling coefficient acting on the particle are obtained with good convergence for various values of the ratio of particle-to-container radii, the relative distance between the centers of the particle and container, the slip coefficients and the relative angular velocity. In the limiting cases, the numerical values of the coupling coefficient for the solid sphere in concentric position with the container and when the particle is near the inner surface of the container are obtained, and the results are in good agreement with the available values in the literature. The variation of the coupling coefficient with respect the parameters considered are tabulated and displayed graphically.