Boundary feedback stabilization of a rotating body-beam system

This paper deals with boundary feedback stabilization of a flexible beam clamped to a rigid body and free at the other end. The system is governed by the beam equation nonlinearly coupled with the dynamical equation of the rigid body. The authors propose a stabilizing boundary feedback law which suppresses the beam vibrations so that the whole structure rotates about a fixed axis with any given small constant angular velocity. The stabilizing feedback law is composed of control torque applied on the rigid body and either boundary control moment or boundary control force (or both of them) at the free end of the beam. It is shown that in any case the beam vibrations are forced to decay exponentially to zero     

Dynamic active impedance for vibration suppression of Large Space Structures

This paper approaches the control of Large Space Structures (LSS) by modulating the impedance of a joint to obtain desired vibration suppression. The suppression of several vibration modes cannot be done efficiently with a constant gain control system, i.e. a constant joint impedance. A dynamic active impedance controller is required and is proposed herein for this purpose. The method is applied to a flexible beam which is modelled by the Euler-Bernoulli equation. The experimental set-up and its operation can emulate a typical slew manoeuvre about a fixed axis. The boundary conditions for the beam in this case are defined for a servomotor at one end and a free condition at the other end. The beam parameters are experimentally identified for the first three modes of vibration. Active impedances are determined separately for the rigid mode and the first three modes of vibration using a pole placement method. The four different active impedances are realized using gain scheduling while transitions between gains follow a cubic polynomial of time. The duration of application of each impedance is determined based on their respective settling time. Preliminary experiments establish the minimum duration for each transition from one active impedance to another in suppressing beam vibrations.     

A new trajectory control of a flexible-link robot based on a distributed-parameter dynamic model

The objective of this study is to develop a new trajectory-tracking control method, free from the so-called spillover instability, for a flexible-link robot. Based on a distributed-parameter dynamic model (a partial differential equation), a new moment-feedback trajectory-tracking control scheme is designed for a one-link flexible robot having a payload at the free end, in which zero geometric boundary conditions at the hub end and non-zero dynamic boundary conditions at the free end are taken into account. The proposed control is then extended to an adaptive scheme to cope with parametric uncertainties. The proposed control is stable for trajectory-tracking control and asymptotically stable at desired goal positions, which is proven by using the Lyapunov stability theorem. In addition, the proposed trajectory-tracking control, based on a distributed-parameter dynamic model, does not have the spillover problem. Furthermore, the control performance is guaranteed regardless of the magnitude of desired angle of rotation, which does not require any additional actuators such as piezoelectric actuators on the link and boundary force or moment actuators at the free end. The effectiveness of the proposed control has been shown by experiments.     

Orientation and stabilization of a flexible beam attached to arigid body: planar motion

The author considers a flexible structure modeled as a rigid body which rotates in inertial space; a light flexible beam is clamped to the rigid body at one end and free one is clamped at the other. It is assumed that the flexible beam performs only planar motion. The equations of motion are obtained by using free body diagrams. Two control problems are posed, namely the orientation and stabilization of the system. It is shown that suitable boundary controls applied to the free end of the beam and suitable control torques applied to the rigid body solve the problems posed above. The proofs are obtained by using the energy of the system as a Lyapunov functional     

Geometrically non-linear free vibrations of clamped-clamped beams with an edge crack

It is well known that a crack in a beam induces a drop in its natural frequencies and affects its modes shapes. This paper is a theoretical investigation of the geometrically non-linear free vibrations of a clamped-clamped beam containing an open crack. The approach uses a semi-analytical model based on an extension of the Rayleigh-Ritz method to non-linear vibrations, which is mainly influenced by the choice of the admissible functions. The general formulation is established using new admissible functions, called “cracked beam functions”, and denoted as “CBF”, which satisfy the natural and geometrical end conditions, as well as the inner boundary conditions at the crack location. Iterative solution of a set of non-linear algebraic equations is obtained numerically, which leads to the basic function contribution coefficients to the displacement response function. Then, an explicit solution is derived and proposed as an alternative procedure, simple and ready to use for engineering applications. Emphasis is made on the backbone curves, i.e. amplitude-frequency dependence, obtained for various crack depth, and the effect of the vibration amplitudes upon the non-linear mode shapes of a cracked beam is examined. The work is restricted to the fundamental mode in order to concentrate on the study of the influence of the crack on the non-linear dynamic response near to the fundamental resonance.     

An investigation of free vibrations of a strain gradient Timoshenko beams with the method of initial values

Investigated herein is the free vibrations of beams based on the strain gradient Timoshenko beam theory with the method of initial values. For the vibration of strain-gradient Timoshenko beam (SGTB), the sixth-order ordinary differential equation and three boundary conditions at each end have been obtained by using the Hamilton principle. The effect of the characteristic length on the frequencies of free vibrations is shown. The frequencies of the SGTB are compared to the frequencies of the strain gradient Euler beam (SGEB), classical Timoshenko beam (CTB) and classical Euler beam (CEB). It has been observed that the high-frequency values of conventional and strain-gradient beams are very different. This result can be used to determine the value of the material characteristic length for a nanobeam for which lengthscale effects are believed to be dominant.

     

Maximum Principle for Optimal Boundary Control of Vibrating Structures with Applications to Beams

A maximum principle is derived for open-loop boundary control of one dimensional structures undergoing transverse vibrations. The optimal control law is obtained using a maximum principle and the applicability of the results to the boundary control of vibrating beams is demonstrated. The method of solution involves the transformation of the original problem into one with homogeneous boundary conditions for a general set of boundary forces and torques. An adjoint variable is introduced and used in the formulation of a Hamiltonian function which in turn leads to the derivation of the maximum principle. The effectiveness of the proposed control mechanism is illustrated numerically and it is shown that the implementation of the optimal boundary control using one force actuator can lead to substantial decrease in the dynamic response of a vibrating beam.     

Well-posedness and exponential stability of two-dimensional vibration model of a boundary-controlled curved beam with tip mass

This paper studies the well-posedness and exponential stability of two-dimensional vibration model of a curved beam with tip mass under linear boundary control. The control task is to stabilise the tangential and radial vibrations, which are coupled due to the beam curvature. To reach the main results of the paper, mathematical analyses based on the semigroup theory and Lyapunov approach are conducted, and it is shown that the proposed closed-loop model holds a unique solution that converges to zero exponentially fast. These analyses are based on a hybrid dynamic model that incorporates two coupled partial differential equations and six boundary conditions, including two ordinary differential equations. Simulation results are used to illustrate the efficacy of the suggested method.     

Modeling and vibration control for a flexible pendulum inverted system based on a PDE observer

This study addresses the problem of trajectory control of a flexible pendulum inverted system on the basis of the partial differential equation (PDE) and ordinary differential equation (ODE) dynamic model. One of the key contributions of this study is that a new model is proposed to simplify the complex system. In addition, this study proposed a nonlinear PDE observer to estimate distributed positions and velocities along flexible pendulum. Singular perturbation method is proposed to solve the coupling system of nonlinear PDE observer. The nonlinear PDE observer is divided into a fast subsystem and a slow subsystem by the use of the singular perturbation method. To stabilise this fast subsystem, a boundary controller is proposed at the free end of the beam. The sliding-mode control method is proposed to design controller for slow subsystems. The asymptotic stability of both the proposed nonlinear PDE observer and controller is validated by theoretical analysis. The results are illustrated by simulation.     

Torsional vibrations of composite bars of variable cross-section by BEM

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of doubly symmetric composite bars of arbitrary variable cross-section. The composite bar consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The beam is subjected to an arbitrarily distributed dynamic twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. A distributed mass model system is employed which leads to the formulation of three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions. These problems are solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced torsional vibrations are considered and numerical examples are presented to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The discrepancy in the analysis of a thin-walled cross-section composite beam employing the BEM after calculating the torsion and warping constants adopting the thin tube theory demonstrates the importance of the proposed procedure even in thin-walled beams, since it approximates better the torsion and warping constants and takes also into account the warping of the walls of the cross-section.     

末端有未知扰动的分布参数柔性机械臂的鲁棒边界控

本文研究在柔性机械臂的末端具有未知扰动的边界控制,以降低机械臂的振动.柔性机械臂的动态特性由偏微分方程表示的分布参数模型描述.在机械臂的末端边界基于Lyapunov直接法进行控制,以调节机械臂的振动.应用本文所提出的边界控制方法,可达到外界干扰下的指数稳定性.所提出的控制方法与系统参数无关,可确保在参数变化下系统具有鲁棒性.最后对所提控制方法的有效性进行了数值模拟.     

Modeling and boundary control of a flexible marine riser coupled with internal fluid dynamics

In this paper, vibration reduction of a flexible marine riser with time-varying internal fluid is studied by using boundary control method and Lyapunov’s direct method. To achieve more accurate and practical riser’s dynamic behavior, the model of marine riser with time-varying internal fluid is modeled by a distributed parameter system (DPS) with partial differential equations (PDEs) and ordinary differential equations (ODEs) involving functions of space and time. The dynamic responses of riser are completely different if the time-varying internal fluid is considered. Boundary control is designed at the top boundary of the riser based on original infinite dimensionality PDEs model and Lyapunov’s direct method to reduce the riser’s vibrations. The uniform boundedness and closed-loop stability are proved based on the proposed boundary control. Simulation results verify the effectiveness of the proposed boundary control.     

Stabilization and disturbance rejection for the beam equation

We consider a system described by the Euler-Bernoulli beam equation. For stabilization, we propose a dynamic boundary controller applied at the free end of the system. The transfer function of the controller is a marginally stable positive real function which may contain poles on the imaginary axis. We then give various asymptotical and exponential stability results. We also consider the disturbance rejection problem     

Boundary control of a circular curved beam using active disturbance rejection control

In this paper, boundary stabilisation of vibration of a circular curved beam in the presence of exogenous disturbances using active disturbance rejection control is addressed. Based on the Euler–Bernoulli beam theory, the vibration of the inextensible curved beam including a tip mass is governed by a sixth-order partial differential equation (PDE) with dynamic boundary conditions. Based on the considered PDE, linear boundary control is introduced that exponentially stabilises the beam without disturbance. Furthermore, to control the beam under exogenous disturbances, the established linear control is enhanced by adding the estimates of disturbances. This is achieved by first designing an extended state observer that estimates the external disturbances using the beam vibration at the controlled end. The stability of the closed-loop system in the sense of Lyapunov is analysed based on the PDE model. Using the Faedo–Galerkin method combined with the compactness argument, it is shown the closed-loop system is well-posed. The efficacy of the suggested method is illustrated using simulation results.     

Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic finite element

In this article, a dynamic finite element formulation for the free vibration analysis of axially loaded bending-torsion coupled beams is presented. Based on the Euler–Bernoulli and St. Venant beam theories, the exact solutions of the differential equations governing the uncoupled vibrations of an axially loaded uniform beam are found. Then, employing these solutions as basis functions, the analytical expressions for uncoupled bending and torsional dynamic shape functions are derived. Exploiting the principle of virtual work, together with the variable approximations based on the resulting shape functions, leads to a single frequency dependent element matrix which has both mass and stiffness properties. The application of the theory is demonstrated by an illustrative example of a bending-torsion coupled beam with cantilever end conditions, for which the influence of axial force on the natural frequencies is studied. The correctness of the theory is confirmed by the published results and numerical checks.     

Topology optimization framework for structures subjected to stationary stochastic dynamic loads

The field of topology optimization has progressed substantially in recent years, with applications varying in terms of the type of structures, boundary conditions, loadings, and materials. Nevertheless, topology optimization of stochastically excited structures has received relatively little attention. Most current approaches replace the dynamic loads with either equivalent static or harmonic loads. In this study, a direct approach to problem is pursued, where the excitation is modeled as a stationary zero-mean filtered white noise. The excitation model is combined with the structural model to form an augmented representation, and the stationary covariances of the structural responses of interest are obtained by solving a Lyapunov equation. An objective function of the optimization scheme is then defined in terms of these stationary covariances. A fast large-scale solver of the Lyapunov equation is implemented for sparse matrices, and an efficient adjoint method is proposed to obtain the sensitivities of the objective function. The proposed topology optimization framework is illustrated for four examples: (i) minimization of the displacement of a mass at the free end of a cantilever beam subjected to a stochastic dynamic base excitation, (ii) minimization of tip displacement of a cantilever beam subjected to a stochastic dynamic tip load, (iii) minimization of tip displacement and acceleration of a cantilever beam subjected to a stochastic dynamic tip load, and (iv) minimization of a plate subjected to multiple stochastic dynamic loads. The results presented herein demonstrate the efficacy of the proposed approach for efficient multi-objective topology optimization of stochastically excited structures, as well as multiple input-multiple output systems.

     

The well-posedness and stability of a beam equation with conjugate variables assigned at the same boundary point

A Euler-Bernoulli beam equation subject to a special boundary feedback is considered. The well-posedness problem of the system proposed by G. Chen is studied. This problem is in sharp contrast to the general principle in applied mathematics that the conjugate variables cannot be assigned simultaneously at the same boundary point. We use the Riesz basis approach in our investigation. It is shown that the system is well-posed in the usual energy state space and that the state trajectories approach the zero eigenspace of the system as time goes to infinity. The relaxation of the applied mathematics principle gives more freedom in the design of boundary control for suppression of vibrations of flexible structures.     

Fault-tolerant control of flexible air-breathing hypersonic vehicles in linear ODE-beam systems

ABSTRACT

This paper addresses the fault-tolerant control issue for a class of flexible air-breathing hypersonic vehicles. Firstly, a longitudinal dynamic model with process faults is established, which contains an ordinary differential equation (ODE) for rigid body, an Euler–Bernoulli beam equation for flexible modes, and a new boundary connection between them; Secondly, a novel fault-tolerant control scheme is proposed to accommodate process faults and suppress vibrations, which relies on the direct Lyapunov method and the bilinear matrix inequalities (BMIs) technique; Thirdly, in order to compute the gain matrices of the fault-tolerant control law, a two-step algorithm is provided to solve the BMI feasibility problem in terms of linear matrix inequality optimisation technique. Finally, the simulation results are provided to illustrate the effectiveness of the theoretical results.     

Nonlinear free bending vibrations of plates

A general solution for the Helmholtz differential equations is obtained in the complex domain and applied to the nonlinear, free, bending vibrations of plates. The analysis is based on the decoupled nonlinear von Karman field equations by Berger assumption for the large deformations of plates. The decoupled differential equation in terms of the deflection function is a fourth order Helmholtz differential equation. Its solution, called the dynamic deflection function, is obtained in the complex domain by means of newly defined first and second kind and modified Bessel functions. The dynamic deflection function can be applied to any plates having any shape and any boundary condition under any arbitrary dynamic loads. For plates with smooth boundary, the parameters of the dynamic deflection function are determined from the boundary conditions of the plates and the initial conditions of the vibrations. The analyses of plates with piece-wise smooth boundaries are obtained on the mapped planes. The nonlinear, free vibration of circular plates are investigated by the dynamic deflection function. The effect of stretching on the natural circular frequencies are illustrated.     

带动态边界的非均匀Timoshenko梁的反馈镇定

对于一端具负载的非均质Timoshenko梁, 研究了其边界反馈镇定问题. 首先提出了一种边界反馈控制方案, 建立了相应的闭环系统的适定性. 然后利用乘子法证明了, 当两个边界反馈控制同时作用于梁的负载端时, 闭环系统是指数稳定的.