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 suppression of a dynamical system to multi-parametric excitations via time-delay absorber

The aim of this work is to control the dynamic system behavior represented by a beam at simultaneous primary and sub-harmonic resonance condition, where the system damage is probable. Control is conducted via time delay absorber to suppress chaotic vibrations. A comprehensive investigation of the effect of the time delay on the control of a beam when subjected to multi- parametric excitation forces is presented. Multiple scale perturbation method is applied to obtain the solution up to the second order approximation. Different resonance cases are reported and studied numerically. Stability of the steady state solution for the selected resonance case is investigated applying Rung-Kutta fourth order method and frequency response equations via Matlab 7.0 and Maple11. Time delay absorber is effective like ordinary one within a specified range of time delay. The delay time is an important factor in selecting the absorber. The effects of the different parameters of the absorber on the system behavior are studied numerically. The reported results are compared with the available published work.     

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.     

Vibration reduction of a three DOF non-linear spring pendulum

The dynamic response of mechanical and civil structures subject to high-amplitude vibration is often dangerous and undesirable. Vibrations and dynamic chaos should be controlled or eliminated in both structures and machines. This can be employed via passive and active control methods. In this paper, a tuned absorber, in the transversally direction, is connected to an externally excited spring–pendulum system (three degree of freedom), subjected to harmonic excitation. The tuned absorber is usually designed to control one frequency at primary resonance where system damage is probable. Active control is also applied to the considered system via negative displacement feedback to change the linear frequency of the system and to shift it away from the resonating one. Also active control is applied to improve the behavior of the spring–pendulum at the primary resonance via negative velocity feedback or its square or cubic value. The multiple time scale perturbation technique is applied throughout. The stability of the system is investigated applying both frequency response function and phase-plane method. The effects of the absorber and different parameters on system behavior are studied numerically. Optimum working conditions of the system are extracted applying both passive and active control methods, to be used in the design of such systems.     

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.     

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.     

Non-linear oscillations of a thin-walled composite beam with shear deformation

A geometrically non-linear theory is used to study the dynamic behavior of a thin-walled composite beam. The model is based on a small strain and large rotation and displacements theory, which is formulated through the adoption of a higher-order displacement field and takes into account shear flexibility (bending and warping shear). In the analysis of a weakly nonlinear continuous system, the Ritz’s method is employed to express the problem in terms of generalized coordinates. Then, perturbation method of multiple scales is applied to the reduced system in order to obtain the equations of amplitude and modulation. In this paper, the non-linear 3D oscillations of a simply-supported beam are examined, considering a cross-section having one symmetry axis. Composite is assumed to be made of symmetric balanced laminates and especially orthotropic laminates. The model, which contains both quadratic and cubic non-linearities, is assumed to be in internal resonance condition. Steady-state solution and their stability are investigated by means of the eigenvalues of the Jacobian matrix. The equilibrium solution is governed by the modal coupling and experience a complex behavior composed by saddle noddle, Hopf and double period bifurcations.     

3:1 Internal resonance of a string-beam coupled system with cubic nonlinearities

The dynamic behavior and chaotic motion of a string-beam coupled system subjected to parametric excitation are investigated. The case of three-to-one internal resonance between the modes of the beam and the string, in the presence of subharmonic resonance for the beam is considered and examined. The method of multiple scales is applied to study the steady-state response and the stability of the string-beam coupled system at resonance conditions. Numerical simulations illustrated that multiple-valued solutions, jump phenomenon, hardening and softening nonlinearities occur in the resonant frequency response curves. The effects of different parameters on system behavior have been studied applying frequency response function. Results are compared to previously published work.     

The resonance stabilization of a class of unstable systems

The Lyapunov stability of the trivial solution of a non-linear system, which, in the first approximation, describes a multifrequency oscillatory process, is investigated. It is shown that a system that is unstable when account is taken of non-linear terms can be made asymptotically stable by tuning it to a fourth-order resonance. Sufficient conditions for asymptotic stability are obtained.     

Second-order approximation of angle-ply composite laminated thin plate under combined excitations

The influence of the quadratic and cubic terms on non-linear dynamic characteristics of the angle-ply composite laminated rectangular plate with parametric and external excitations is investigated. The method of multiple time scale perturbation is applied to solve the non-linear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted and investigated at this approximation order. Two cases of the sub-harmonic resonances cases (Ω₂ ? 2ω₁ and Ω₂ ? 2ω₂) in the presence of 1:2 internal resonance ω₂ ? 2ω₁ are considered. The stability of the system is investigated using both frequency response equations and phase-plane method. It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system as they represent some of the worst behavior of the system. Such cases should be avoided as working conditions for the system. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.     

Pitfalls in the frequency response represented onto Polynomial Chaos for random SDOF mechanical systems

Uncertainties are present in the modeling of dynamical systems and they must be taken into account to improve the prediction of the models. It is very important to understand how they propagate and how random systems behave. This study aims at pointing out the somehow complex behavior of the structural response of stochastic dynamical systems and consequently the difficulty to represent this behavior using spectral approaches. The main objective is to find numerically the probability density function (PDF) of the response of a random linear mechanical systems. Since it is found that difficulties can occur even for a single-degree-of-freedom system when only the stiffness is random, this work focuses on this application to test several methods. Polynomial Chaos performance is first investigated for the propagation of uncertainties in several situations of stiffness variances for a damped single-degree-of-freedom system. For some specific conditions of damping and stiffness variances, it is found that numerical difficulties occur for the standard polynomial bases near the resonant frequency, where it is generally observed that the shape of the system response PDFs presents multimodality. Strategies to build enhanced bases are then proposed and investigated with varying degrees of success. Finally, a multi-element approach is used in order to gain robustness.     

Nonlinear study of a rotor–AMB system under simultaneous primary-internal resonance

A rotor–active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic non-linearities under multi-parametric excitations is studied and solved. The method of multiple scales is applied to analyze the response of two modes of a rotor–AMB system with multi-parametric excitations and time-varying stiffness near the simultaneous primary and internal resonance. The stability of the steady state solution for that resonance is determined and studied using Rung–Kutta method of fourth order. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening and softening non-linearities and chaos in the second mode of the system. The effects of the different parameters on the steady state solutions are investigated and discussed also. A comparison to published work is reported.     

Near-resonant motions in systems with random perturbations

The effect of random perturbations on near-resonant motions in non-linear oscillatory systems is investigated. It is assumed that the equations of motion of the system can be reduced to standard form with a small parameter ϵ, and that an isolated primary resonance exists in the unperturbed system [1]. The behaviour of the perturbed system in the ϵ-neighbourhood of the resonance surface is considered and an effect analogous to deterministic “capture in resonance” [1] in an asymptotically long time interval is investigated.     

On the non-ideal vibrating structures

In this work, analysis of the response to vibrations of a small building equipped with an electromechanical vibration absorber is investigated. The case of harmonic excitation with constant or time dependent frequencies is considered. The interaction between the structure and the energy source is analyzed via the Sommerfeld effect inside the resonance region. The resonance capture and the vibration reduction are displayed by the time history displacement of the last story. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic behavior of an AMB supported rotor subject to harmonic excitation

This paper presents a study of the non-linear response of a simple rigid disk-rotor, supported by active magnetic bearings (AMB), without gyroscopic effects. The case of primary resonance is examined under multi-excitation forces. The rotating shaft is described by a coupled second order non-linear ordinary differential equations. Approximate solutions are sought applying the method of multiple scales. Numerical simulations are carried out to illustrate the steady-state response and the stability of the solutions for various parameters using the frequency response function method. It is shown that the system parameters have different effects on the non-linear response of the rotor. For steady-state response, however, multiple-valued solutions and jump phenomenon occur. Results are compared to previously published work.     

A connection between multi-linear and Volterra systems

The Volterra system is a non-linear system with the structure of the Volterra series. The Volterra series is attractive from the system-theoretic point of view, since it enables to obtain the output of a class of non-linear systems in terms of the input explicitly rather than involving input-output coupling terms and allows substantial simplifications for the numerical simulation. The Volterra system allows to derive the stability condition as well, i.e. obtain a bound on the output for a given bound on the input function, especially for the bilinear system. The bilinear system possesses the Volterra series. This paper derives the Volterra series formalism from the multi-linear system involving the coupling term attributed to (k-1)th order non-linearity and output function, where 1<k. The bilinear system becomes a special case of the non-linear problem of concern here. The Volterra series formalism of this paper is derived using the discrete counterpart of the phase space analysis for the non-linear non-autonomous system. The main result of the paper, i.e the Volterra series formalism of the multi-linear system of concern here, is somewhat more general, since the Volterra series representations for bilinear and tri-linear systems, etc. can be obtained as its special cases.     

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.     

Evolution of passive movement in advective environments: General boundary condition

In a previous work [16], Lou et al. studied a Lotka–Volterra competition–diffusion–advection system, where two species are supposed to differ only in their advection rates and the environment is assumed to be spatially homogeneous and closed (no-flux boundary condition), and showed that weaker advective movements are more beneficial for species to win the competition. In this paper, we aim to extend this result to a more general situation, where the environmental heterogeneity is taken into account and the boundary condition at the downstream end becomes very flexible including the standard Dirichlet, Neumann and Robin type conditions as special cases. Our main approaches are to exclude the existence of co-existence (positive) steady state and to provide a clear picture on the stability of semi-trivial steady states, where we introduced new ideas and techniques to overcome the emerging difficulties. Based on these two aspects and the theory of abstract competitive systems, we achieve a complete understanding on the global dynamics.     

Some sufficient conditions for the non-negativity preservation property in the discrete heat conduction model

In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments.     

Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves

The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition.