Vibration suppression of non-linear system via non-linear absorber

In this paper, the coupled non-linear differential equations of the non-linear dynamical two-degree-of-freedom vibrating system including quadratic and cubic non-linearities are studied. The system consists of the main system and the absorber. The absorber is used to control the main system vibrations when subjected to multi-external excitation forces at simultaneous primary and internal resonance. This system represents many applications in machine tools, ultrasonic cutting process, etc. The method of multiple scales perturbation technique (MSPT) is applied throughout to determine the solution up to third order approximations. The different resonance cases are reported and studied numerically. Stability is studied applying frequency response functions. The effects of different parameters of the system are studied numerically. Optimum working conditions for the absorber where obtained at internal resonance ratio 1:3. This means smaller mass for the absorber which solves the problem of space limitation. A comparison is made with the available published work.     

Vibration reduction in a 2DOF twin-tail system to parametric excitations

The use of active feedback control strategy is a common way to stabilize and control dangerous vibrations in vibrating systems and structures, such as bridges, highways, buildings, space and aircrafts. These structures are distributed-parameter systems. Unfortunately, the existing vibrations control techniques, even for these simplified models, are fraught with numerical difficulties and engineering limitations. In this paper, a negative velocity feedback is added to the dynamical system of twin-tail aircraft, which is represented by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities. The system describes the vibration of an aircraft tail subjected to multi-parametric excitation forces. The method of multiple time scale perturbation is applied to solve the nonlinear differential equations and obtain approximate solutions up to the third order approximations. The stability of the system is investigated applying frequency response equations. The effects of the different parameters are studied numerically. Some different resonance cases are investigated. A comparison is made with the available published work.     

A Modal Analysis of Resonance during the Whole-Body Vibration

The mechanism of resonance of a damping system of multi‐degrees of freedom such as the human body and the dependence of resonance on system parameters, particularly on the damping level, are studied in terms of detailed mathematical solutions of both the whole‐body vibrations and the eigen modes for a simple model. It is revealed that resonance would only occur near the eigen frequencies of neutral modes for which the complex eigen frequencies of the corresponding damping modes for the given damping level of the system have not moved far from the starting point (damping‐free case) along the corresponding tracks in the plane of complex eigen frequency yet. The major resonance would occur near the eigen frequency of the neutral mode where the modulus of the characteristic function of the system has the strongest, i.e., the deepest and sharpest, local minimum. For the present model, this neutral mode is the lowest neutral mode. It is found that the resonance and eigen frequencies increase with the stiffness of muscles and decrease with the body mass, with the portion of wobbling mass in the upper body, and with the portion of upper body mass in the whole body. Both the modal analysis and the analysis of the whole‐body vibration show that the phase differences among different parts of the system are still small at the unique or the lowest resonance frequency and increase dramatically only when the frequency of the vibrating source goes beyond the resonance frequency. Thus, some effects of body vibrations, e.g., internal loads, may reach their maximum not at the resonance frequency, but at a frequency somewhat higher than the resonance frequency. This may account for the fact that the frequency ranges for abdominal pain and for lumbosacral pain caused by body vibrations are not exactly the same as the frequency range for major body resonance but shifted to somewhat higher frequency ranges. It is therefore suggested that the frequency used for strength training in terms of vibrating devices should be above 20 Hz in order to avoid not only the major resonance but also the maximal internal loads.     

Triple scale analysis of periodic solutions and resonance of some asymmetric non linear vibrating systems

We consider small solutions of a vibrating mechanical system with smooth non-linearities for which we provide an approximate solution by using a triple scale analysis; a rigorous proof of convergence of the triple scale method is included; for the forced response, a stability result is needed in order to prove convergence in a neighbourhood of a primary resonance. The amplitude of the response with respect to the frequency forcing is described and it is related to the frequency of a free periodic vibration.     

多个同频摇荡剖面引起的水面波辐射

在内场中使用简单Green函数的边界元方法与外场的速度势特征函数展开式相结合,用于求解多个同频摇荡剖面引起的水面波辐射问题的频域解·　方法适用于外场为定深的水域以及内场的复杂边界条件,各剖面的摇荡模态、幅值和相位可以互不相同·　利用摄动展开完整地求解了流场的二阶速度势和各个剖面所受的一、二阶水动力·　与单个剖面的情况相比,数值结果证实了多个剖面辐射引起的诸如水波共振和负附加质量等水动力干扰现象,这对于多体结构的锚泊系统和其它海洋工程设施的设计是很重要的·     

Bifurcation of periodic solutions of a system close to a Lyapunov system

Bifurcation of 2π-periodic solutions (2π-ps) of a system of second-order differential equations close to a Lyapunov system is investigated. The case of principal resonance, when an eigenfrequency of the linear oscillations of the unperturbed system is close to the frequency of the perturbing impulse, is considered. It is shown that, at certain values of the problem parameters, bifurcation of the 2π-ps that are generated from an equilibrium position, occurs. A constructive method is proposed for finding the bifurcation curve, as well as 2π-ps on it. The examples considered are bifurcation of 2π-ps in the problem of the oscillations of a mathematical pendulum with a horizontally vibrating suspension point, and in the problem of the planar oscillations of an artificial satellite in a weakly elliptical orbit. The bifurcation curves for these examples are constructed and the corresponding 2π-ps are found.     

Optimal Control of a Constrained Bilinear Dynamic System

In this paper, an optimal feedback, for a free vibrating semi-active controlled plant, is derived. The problem is represented as a constrained optimal control problem of a single input, free vibrating bilinear system, and a quadratic performance index. It is solved by using Krotov’s method and to this end, a novel sequence of Krotov functions that suits the addressed problem, is derived. The solution is arranged as an algorithm, which requires solving the states equation and a differential Lyapunov equation in each iteration. An outline of the proof for the algorithm convergence is provided. Emphasis is given on semi-active control design for stable free vibrating plants with a single control input. It is shown that a control force, derived by the proposed technique, obeys the physical constraint related with semi-active actuator force without the need of any arbitrary signal clipping. The control efficiency is demonstrated with a numerical example.     

Optimal boundary control of dynamics responses of piezo actuating micro-beams

Optimal control theory is formulated and applied to damp out the vibrations of micro-beams where the control action is implemented using piezoceramic actuators. The use of piezoceramic actuators such as PZT in vibration control is preferable because of their large bandwidth, their mechanical simplicity and their mechanical power to produce controlling forces. The objective function is specified as a weighted quadratic functional of the dynamic responses of the micro-beam which is to be minimized at a specified terminal time using continuous piezoelectric actuators. The expenditure of the control forces is included in the objective function as a penalty term. The optimal control law for the micro-beam is derived using a maximum principle developed by Sloss et al. [J.M. Sloss, J.C. Bruch Jr., I.S. Sadek, S. Adali, Maximum principle for optimal boundary control of vibrating structures with applications to beams, Dynamics and Control: An International Journal 8 (1998) 355–375; J.M. Sloss, I.S. Sadek, J.C. Bruch Jr., S. Adali, Optimal control of structural dynamic systems in one space dimension using a maximum principle, Journal of Vibration and Control 11 (2005) 245–261] for one-dimensional structures where the control functions appear in the boundary conditions in the form of moments. The derived maximum principle involves a Hamiltonian expressed in terms of an adjoint variable as well as admissible control functions. The state and adjoint variables are linked by terminal conditions leading to a boundary-initial-terminal value problem. The explicit solution of the problem is developed for the micro-beam using eigenfunction expansions of the state and adjoint variables. The numerical results are given to assess the effectiveness and the capabilities of piezo actuation by means of moments to damp out the vibration of the micro-beam with a minimum level of voltage applied on the piezo actuators.     

Effects of Damping Mechanisms in Free and Forced Vibrations of Some Structural Members

For continuous vibrating systems, such as bars and beams, end-mounted in the environment, knowledge about the mass, damping and stiffness properties of the resonating environment is important for analyzing free and forced vibrations of such structural members which are also damped themselves. To finally get an identification of the clamping parameters, examinations of both vibrating structural members for various restraint conditions and dynamic interaction with viscoelastic halfspaces, etc., are required. As a first step, longitudinal bar vibrations are studied in detail. The method of separation of variables combined with the reformatted orthogonality relation, and numerical integration is applied for calculating the free and forced oscillations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

1:2 and 1:3 internal resonance active absorber for non-linear vibrating system

Vibration and dynamic chaos should be controlled in either structures or machines. An active vibration absorber for suppressing the vibration of the non-linear plant when subjected to external and parametric excitations is studied in the presence of one-to-two and one-to-three internal resonance. The main attention is focused on the study of the active control and stability of two systems, which can be used to reduce vibrations due to rotor blade flapping motion. The method of multiple scale perturbation technique is applied to determine four first-order non-linear ordinary differential equations that govern the modulation of the amplitudes and phases in the presence of internal resonance of the two systems with quadratic and cubic order of control. These equations are used to determine the steady state solutions and their stability. The stability study of non-linear periodic solution for two cases (1:2 and 1:3 internal resonance) and the stability of the obtained numerical solution are investigated using frequency, force-response curves and phase-plane method. Also, effects of some parameters on the steady state solution of the vibrating system are investigated and reported in this paper. Variation of some parameters leads to the bending of the frequency, force-response curves and hence to the jump phenomenon occurrence. The reported results are compared to the available published work.     

Maximum principle for optimal control of distributed-parameter systems with singular mass matrix

A maximum principle for the open-loop optimal control of a vibrating system relative to a given convex index of performance is investigated. Though maximum principles have been studied by many people (see, e.g., Refs. 1–5), the principle derived in this paper is of particular use for control problems involving mechanical structures. The state variable satisfies general initial conditions as well as a self-adjoint system of partial differential equations together with a homogeneous system of boundary conditions. The mass matrix is diagonal, constant, and singular, and the viscous damping matrix is diagonal. The maximum principle relates the optimal control with the solution of the homogeneous adjoint equation in which terminal conditions are prescribed in terms of the terminal values of the optimal state variable. An application of this theory to a structural vibrating system is given in a companion paper (Ref. 6).     

On the numerical solution of certain boundary control problems for vibrating media in one space-dimension

This paper is concerned with the numerical solution of control problems which consist of minimizing certain quadratic functionals depending on control functions in L ₂[0,1] for some given time T > 0 and bounded with respect to the maximum norm. These control functions act upon the boundary conditions of a vibrating system in one space-dimension which is governed by a wave equation of spatial order 2n They are to be chosen in such a way that a given initial state of vibration at time zero is transferred into the state of rest. This requirement can be expressed by an infinite system of moment equations to be satisfied by the control functions

The control problem is approximated by replacing this infinite system by finitely many, say N, equations (truncation) and by choosing piecewise constant functions as controls (discretization). The resulting problem is a quadratic optimization problem which is solved very efficiently by a multiplier method

Convergence of the solutions of the approximating problems to the solution of the control problem, as N tends to infinity and the discretization is infinitely refined, is shown under mild assumptions. Numerical results are presented for a vibrating beam     

Shaking table experimental study on the effectiveness of polymer bearings for seismic isolation of structures

Seismic isolation has been recognised to be a very effective way of protecting structures from damage during earthquakes. It allows us to extend the natural period of the structure and therefore avoid resonance with the ground motion. Moreover, by increasing damping in the isolation devices, more energy can be dissipated and thus the structural response can be further reduced. The aim of this paper is to show the results of the study focused on verification of the applicability of the innovative method of seismic isolation by installing bearings made of a polymer mass, which is an especially designed flexible elastoplastic two-component grout based on polyurethane resin. In the study, a model structure was tested experimentally on a shaking table under earthquake excitation. First, the structure was fixed directly to the table and its response was recorded. Then, the response of the structure with polymer bearings installed at its base was observed and both responses were compared. The results of the study show that equipping the structure with the polymer bearings can considerably reduce the structural response under earthquake excitation. The innovative method considered in the study has been verified to be an effective seismic isolation technique. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of critical influential speed for moving mass problems on beams

A traveling mass due to its mass inertia has significant effects on the dynamic response of the structures. According to recent developments in structural materials and constructional technologies, the structures are likely to be affected by sudden changes of masses and substructure elements, in which the inertia effect of a moving mass is not negligible. The transverse inertia effects have been a topic of interest in bridge dynamics, design of railway tracks, guide way systems and other engineering applications such as modern high-speed precision machinery process. In this study an analytical–numerical method is presented which can be used to determine the dynamic response of beams carrying a moving mass, with various boundary conditions. It has been shown that the Coriolis acceleration, associated with the moving mass as it traverses along the vibrating beam shall be considered as well. Influences regarding the speed of the moving mass on the dynamic response of beams with various boundary conditions were also investigated. Results illustrated that the speed of a moving mass has direct influence on the entire structural dynamic response, depending on its boundary conditions. Critical influential speeds in the moving mass problems were introduced and obtained in numerical examples for various BC’s.     

Stability study and control of helicopter blade flapping vibrations

The response of a two-degree-of-freedom, controlled, autoparametric system to harmonic excitations is studied and solved. The objective of this research is to investigate the effect of linear absorber on the vibrating system and the saturation control of a linear absorber to reduce vibrations due to rotor blade flapping motion. The method of multiple scale perturbation technique is applied to obtain the periodic response equation near the primary resonance in the presence of internal resonance of the system. The stability of the obtained numerical solution is investigated using both phase plane methods and frequency response equations. Variation of some parameters leads to the bending of the frequency response curves and hence to the jump phenomenon occurrence. The reported results are compared to the available published work.     

On optimal design of vibrating structures

Optimal design problems of linearly elastic vibrating structural members have been formulated in two ways. One is to minimize the total mass holding the frequency fixed; the other is to maximize the fundamental frequency holding the total mass fixed. Generally, these two formulations are equivalent and lead to the same solution. It is shown in this work that the equivalence is lost when the design variable (the specific stiffness) appears linearly in Rayleigh's quotient and when there is no nonstructural mass. The maximum-frequency formulation then is a normal Lagrange problem, whereas the minimum-mass problem is abnormal. The lack of recognition of this can lead to incorrect conclusions, particularly concerning existence of solutions. It is shown that existence depends directly on the boundary conditions and, when a sandwich beam has a free end, a solution to the maximum-frequency problem does not exist.     

On boundary controllability of one-dimensional vibrating systems

In this paper boundary controllability of one-dimensional vibrating system such as the vibrating string or the vibrating beam is studied. In particular we are concerned with the question whether it is possible to transfer a given initial state of vibration into rest within a given time such that the system stays in rest when the control is turned off. This problem is rephrased as a typical trigonometric moment problem which is solved within the framework of an abstract moment problem in a Hilbert space. The results of null-controllability which are obtained are substantially based on classical results of Ingham and Redheffer concerning trigonometric inequalities and incompleteness of certain sequences of trigonometric functions, respectively. The representation of the general statements follows closely the lines of a paper of Russell. Besides a special case is treated where explicit representations of boundary controls can be given that transfer the system to a permanent rest position. This special case includes amplitude boundary control of the vibrating string and the freely supported beam.     

A Coupled FE-BE Analysis of Smart Lightweight Structures for Active Noise and Vibration Control

Over the past years much research and development has been done in the area of active control in order to improve the acoustical and vibrational properties of thin–walled lightweight structures. An efficient technique for actively reducing the structural vibration and sound radiation is the application of smart structures. In smart structures piezoelectric materials are often used as actuators and sensors. The design of smart structures requires fast and reliable simulation tools. Therefore, the purpose of this paper is to present a coupled finite element–boundary element formulation, which enables the modeling of piezoelectric smart lightweight structures. The paper describes the theoretical background of the coupled approach in which the finite element method (FEM) is applied for the modeling of the passive vibrating shell structure as well as the surface attached piezoelectric actuators and sensors. The boundary element method (BEM) is used to characterize the corresponding sound field. In order to derive a coupled FE–BE formulation additional coupling conditions are introduced at the fluid–structure interface. Since the resulting overall model contains a large number of degrees of freedom, the mode superposition method is employed to reduce the size of the FE submodel. To validate the accuracy of the proposed approach, numerical simulations are carried out in the frequency domain and the results are compared with analytical reference solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibration reduction in ultrasonic machine to external and tuned excitation forces

The dynamic response of mechanical and civil structures subject to high-amplitude vibration is often dangerous and undesirable. Sometimes controlled vibration is desirable as in ultrasonic machining (USM). Ultrasonic machining (USM) is the removal of material by the abrading action of grit-loaded liquid slurry circulating between the workpiece and a tool vibrating perpendicular to the workface at a frequency above the audible range. A high-frequency power source activates a stack of magnetostrictive material, which produces a low-amplitude vibration of the toolholder. This motion is transmitted under light pressure to the slurry, which abrades the workpiece into a conjugate image of the tool form. This can be achieved via passive and active control methods. In this paper, multi-tool techniques are used in the ultrasonic machining via reducing the vibration in the tool holder and providing reasonable amplitudes for the tools represented by the absorbers. The coupling of the tool holder and absorbers simulating ultrasonic cutting process are investigated. This leads to a multi-degree-of-freedom system subject to external and tuned excitation forces. Multiple scale perturbation method is applied to obtain the solution up to the second order approximation. Different resonance cases are reported and studied numerically. The stability of the system is investigated applying both phase-plane and frequency response techniques. The effects of the different parameters of the absorbers on the system behavior are studied numerically. Comparison with the available published work is reported.