一个复合系统边界反馈的Riesz基性质

该文考虑一端固定 ,一端具负荷的梁的振动问题 .证明了线性反馈的闭环系统是一个 Riesz谱系统 ,即系统存在一列广义本征函数列构成状态空间的 Riesz基 .从而系统的谱确定增长条件成立 .在此过程中 ,简单的导出了系统本征值的渐近展开式 .并因此推论出系统的指数稳定性的条件     

Control of a viscoelastic translational Euler–Bernoulli beam

In this paper, we study a cantilevered Euler–Bernoulli beam fixed to a base in a translational motion at one end and to a tip mass at its free end. The beam is subject to undesirable vibrations, and it is made of a viscoelastic material that permits a certain weak damping. By applying a control force at the base, we shall attenuate these vibrations in a fast manner. In fact, we establish the exponential stability of the system. Our method is based on the multiplier technique. Copyright © 2016 John Wiley & Sons, Ltd.     

Mathematical modelling and dynamic response of a multi-straight-line path tracing flexible robot manipulator with rotating-prismatic joint

In this study, mathematical modelling and dynamic response of a flexible robot manipulator with rotating-prismatic joint are investigated. The tip end of the flexible robot manipulator traces a multi-straight-line path under the action of an external driving torque and an axial force. Considered robot manipulator consists of a rotating prismatic joint and a sliding flexible arm with a tip mass. Flexible arm is assumed to be an Euler–Bernoulli beam carrying an end-mass. Equations of motion of the flexible manipulator are obtained by using Lagrange’s equation of motion. Effect of rotary inertia, axial shortening and gravitation is considered in the analysis. Equations of motion are solved by using fourth order Runge–Kutta method. Numerical simulations obtained by using a developed computer program are presented and physical trend of the results are discussed.     

Simulation of the transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction using integral transforms

The main purpose of the current work is to employ an integral transform approach based on eigenfunction expansion and on an implicit filter scheme in order to solve the governing equations for the transverse vibrations of a cantilever beam clamped at one end and with an eccentric tip mass in the axial direction at the other end. Numerical results are obtained for both the undamped and damped natural frequencies of the system, as well as for its transverse displacement due to arbitrarily time-varying load and imposed displacement at the clamped end. The numerical results reported in the current work are highly accurate and new in the literature. New exact results are also provided for the transient displacement and its higher-order spatial derivatives to allow computation of bending stresses and strains. The relative merits of the proposed approach are finally pointed out.     

Uniform stability for the Timoshenko beam with tip load

In this paper we consider a hybrid elastic model consisting of a Timoshenko beam and a tip load at the free end of the beam. We show that uniform stabilization of the model which includes the rotary inertia of the tip load can be obtained when feedback boundary moment and force controls are applied at the point of contact between the beam and the tip load. However, in the presence of the load stabilization is “slower” and subject to a restriction on the boundary data at the free end of the beam.     

A criterion for the growth of cracks with bonds in the end zone

A fracture criterion which takes account of the work done in the deformation of bonds in the end zone of a crack is proposed for analysing the quasistatic growth of a crack with bonds in the end zone. The energy condition that the deformation energy release rate at the crack tip is equal to the rate of deformation energy consumption by the bonds in the end zone of the crack (the first fracture condition) corresponds to the state of limit equilibrium of the crack tip. The rupture of bonds at the trailing edge of the end zone is determined by the condition for their limiting traction (the second fracture condition). Starting from these two conditions, the processes of subcritical and quasistatic crack growth are considered for the case of a rectilinear crack at interface of materials and the two basic fracture parameters, the critical external load and the size of the end zone of the crack in the state of limit equilibrium, are determined. Analytical expressions are obtained for the deformation energy release rate at the crack tip and the rate of deformation energy consumption by the bonds and, also, the dependences of the critical external load and size of the end zone of the crack on the crack length in the case of a rectilinear crack in a homogeneous body with bond tractions which are constant and independent of the external load. The limit cases of a crack which is filled with bonds and a crack with a short end zone are considered.     

A Combination of Flatness-Based Tracking Control with Passivity-Based Control for a Certain Class of Infinite-Dimensional Systems

This contribution is devoted to the infinite-dimensional control design for a certain class of infinite-dimensional systems. As first example a piezoelectric cantilever with a tip mass is considered. The control objective is to provide two independently controllable degrees-of-freedom for the tip mass in form of the tip position and the tip angle. The control concept being proposed consists of an open-loop flatness-based tracking controller and a linear dynamic feedback controller in order to asymptotically stabilize the closed-loop error system. A similar concept is then applied to a second example, a gantry crane system with heavy chains and a payload. Thereby, the knowledge of the energy flows into and within the system is exploited to derive a stabilizing controller of the error system by means of the integrator backstepping method. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The motion of a particle on a light viscoelastic bar: asymptotic analysis of its quasilinear parabolic–hyperbolic equation

We study the large longitudinal motion of a nonlinearly viscoelastic bar with one end fixed and the other end attached to a heavy particle. This problem is a precise continuum-mechanical analog of the basic discrete mechanical problem of the motion of a particle on a (massless) spring. This motion is governed by an initial-boundary-value problem for a class of third-order quasilinear parabolic–hyperbolic partial differential equations subject to a nonstandard boundary condition, which is the equation of motion of the particle. The ratio of the mass of the bar to that of the tip mass is taken to be a small parameter ε. We prove that this problem has a unique globally defined solution that admits a valid asymptotic expansion, including an initial-layer expansion, in powers of ε for ε near 0. The validity of the expansion gives a precise meaning to the solution of the reduced problem, obtained by setting ε=0, which curiously is seldom governed by the expected ordinary differential equation. The fundamental constitutive hypothesis that the tension be a uniformly monotone function of the strain rate plays a critical role in a delicate proof that each term of the initial-layer expansion decays exponentially in time. These results depend on new decay estimates for the solution of quasilinear parabolic equations.     

Optimal design in axial motion for several frequency constraints

Governing equations for the optimal design of a rod with tip mass subject to several constraints on natural frequency are developed. The relation of this mass distribution to the globally optimum design for lowest frequency is discussed. A numerical example for two frequencies is presented. Some continuity results concerning the optimal mass distribution are presented.     

末端质量作用下变长度轴向运动梁的振动特性

对带集中质量,变长度(或速度)轴向运动梁的振动特性采用两种精确方法求解.首先,对变长度轴向运动Euler(欧拉)梁横向自由振动方程进行化简,通过复模态分析得到本征方程,并在有集中质量的边界条件下得到频率方程,用数值方法求解固有频率和模态函数.然后,采用有限元方法建立运动梁自由振动的方程,求解矩阵方程得到复特征值和复特征向量,结合形函数得到复模态位移.最后,将两种方法的计算结果进行了分析和对比.数值算例的结果表明:不同的轴向运动速度和集中质量对变长度轴向运动梁的振动特性有显著影响,两种计算方法的结果接近且均有效.     

考虑大变形的刚-柔耦合旋转智能结构动力学分析

基于Hamilton原理对带端部质量的刚柔耦合旋转智能结构建立了耦合的非线性动力学模型．根据一阶近似耦合（FOAC）模型理论,通过有限元方法,得到了系统的有限维模型．模型中考虑了轴向、横向位移和转动角度的非线性几何效应,以及压电材料和结构的大变形及离心刚化效应．在有限元模型的基础上,建立了3种实际系统模型方程,分别是无压电层的结构,有压电层开环状态和闭环状态．最后基于简化模型的仿真结果显示出有端部质量和没有端部质量的差异,智能结构梁在闭环和开环的差异,高速旋转梁的离心作用及结构外加电载荷的动力响应．     

Thermomechanical behaviour of a damageable beam in contact with two stops

A model for the thermomechanical behaviour of a beam which allows for the general evolution of material damage is presented and investigated. One end of the beam is fixed while the other is constrained to move between two stops. The contact of the free tip with the stops is modelled by the normal compliance condition. The thermal interaction between the stops and the free tip is described by a heat exchange condition where the heat transfer coefficient is a general function of the gaps between the tip and the stops. The effects on the mechanical properties of the material due to crack expansion are described by a damage field, which measures the decrease in the load-bearing capacity of the material. The damage evolves as a constrained diffusion process in which the microcracks that develop may grow or disappear. The mathematical model consists of a coupled system of energy--elasticity equations together with a nonlinear parabolic inclusion for the damage field. The existence of a local solution is established using truncation, penalization, and a priori estimates.     

环形梁塑性动力响应特性研究

本文采用刚塑性假设,对环形悬臂梁在自由端受一刚性质量径向冲击时的塑性动力响应问题进行了分析与研究.给出了质点响应速度与移动塑性铰位置间的精确表达式,并由此发现了环形梁塑性动力响应的一些特殊性质.     

Exact mode shapes for a continuous tether

The vibrational problem of a tether modeled as a cable terminated by two end masses is considered (dumbbell configuration). For an inextensible cable the mode shapes are combinations of Legendre functions. The spacing of the frequencies depends on the mass ratios of the end masses and the cable mass. For the extensible cable, the exact vibrational problem is set up. The conditions under which approximations by more simple models are valid are derived.     

Dynamic modelling of a rigid-flexible manipulator for constrained motion task control

A mathematical model governing the dynamics of a constrained rigid-flexible manipulator moving in a horizontal plane is derived using Hamilton's principle. A new variable is introduced before the procedure of modal expansion in order to convert the non-homogeneous boundary condition into a homogeneous one. The static tip deflection of the flexible link is allowed in order to maintain the contact force between the end effector and the constrained path and this tip deflection is considered in both the inverse kinematics and the order reduction procedures. The state vector of the proposed controller consists of joint angle of the rigid link, its derivative and integral, the first deflection mode and its derivative, and the integral of contact force. A multivariable controller is proposed for the simultaneous motion and force control of the manipulator. The controller consists of a feedforward term which contributes the torque for the expected joint angles and the contact force, and a feedback term with the time varying optimal gains obtained from the Matrix Riccati equation. Computer simulation results show that this proposed controller is capable of performing the straight line tracking task satisfactorily under four different conditions.     

Riesz Basis Generation of Abstract Second-Order Partial Differential Equation Systems with General Non-Separated Boundary Conditions

Riesz basis analysis for a class of general second-order partial differential equation systems with nonseparated boundary conditions is conducted. Using the modern spectral analysis approach for parameterized ordinary differential operators, it is shown that the Riesz basis property holds for the general system if its associated characteristic equation is strongly regular. The Riesz basis property can then be readily established in a unified manner for many one-dimensional second-order systems such as linear string and beam equations with collocated or noncollocated boundary feedbacks and tip mass attached systems. Three demonstrative examples are presented.     

Degenerate cases of stability loss of an elastic fluid-conveying tube

The two-dimensional non-linear dynamics of a liquid-filled tube is considered. The tube is clamped at the upper end, a point mass is fixed to its free lower end and laterally it is supported by two springs. The uniform flow velocity of the fluid, the end mass, the spring constant and the vertical position of the springs are considered as the distinguished parameters of the problem. A linear stability analysis shows that the (degenerate) case of a Takens-Bogdanov-Hopf bifurcation exists, which is associated with a high frequency flutter movement superimposed on a low frequency flutter around a statically buckled state of the tube. We account for this degenerate case by indicating the parameter regime necessary for its occurence and and give the bifurcation diagram for the trivial equilibrium position of the tube. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Self-similar solution of a tensile crack problem in a coupled formulation

Using a self-similar variables, an asymptotic investigation is carried out into the stress fields and the rates of creep deformations and degree of damage close to the tip of a tensile crack under creep conditions in a coupled (creep - damage) plane formulation of the problem. It is shown that a domain of completely damaged material (DCDM) exists close to the crack tip. The geometry of this domain is determined for different values of the material parameters appearing in the constitutive relations of the Norton power law in the theory of steady-state creep and a kinetic equation which postulates a power law for the damage accumulation. It is shown that, if the boundary condition at the point at infinity is formulated as the condition of asymptotic approximation to the Hutchinson–Rice-Rosengren solution [Hutchinson JW. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 1968;16(1):13–31; Rice JR, Rosengren GF. Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids. 1968;16(1):1–12], then the boundaries of the DCDM, which are defined by means of binomial and trinomial expansions of the continuity parameter, are substantially different with respect to their dimension and shape. A new asymptotic of the for stress field, which determines the geometry of the DCDM and leads to close configurations of the DCDM constructed using binomial and trinomial asymptotic expansions of the continuity parameter, are established by an asymptotic analysis and a numerical solution of the non-linear eigenvalue problem obtained.