Showing papers in "Nonlinear Dynamics in 2007"
TL;DR: In this paper, the stability of an n-dimensional linear fractional differential equation with time delays was studied, where the delay matrix is defined in (R+n×n).
Abstract: In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R+)n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined.
748 citations
TL;DR: In this paper, the pull-in instability in microelectromechanical (MEMS) resonators was studied and the authors proposed a low-voltage MEMS RF switch actuated with a combined DC and AC loading, which uses a voltage much lower than the traditionally used DC voltage.
Abstract: We study the pull-in instability in microelectromechanical (MEMS) resonators and find that characteristics of the pull-in phenomenon in the presence of AC loads differ from those under purely DC loads. We analyze this phenomenon, dubbed dynamic pull-in, and formulate safety criteria for the design of MEMS resonant sensors and filters excited near one of their natural frequencies. We also utilize this phenomenon to design a low-voltage MEMS RF switch actuated with a combined DC and AC loading. The new switch uses a voltage much lower than the traditionally used DC voltage. Either the frequency or the amplitude of the AC loading can be adjusted to reduce the driving voltage and switching time. The new actuation method has the potential of solving the problem of high driving voltages of RF MEMS switches.
421 citations
TL;DR: In this paper, the homotopy analysis method (HAM) is applied to obtain the soliton solution of the fifth-order KdV equation, which contains the auxiliary parameter ℏ, which provides a simple way to adjust and control the convergence region of series solution.
Abstract: An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the fifth-order KdV equation. The homotopy analysis method (HAM) provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of series solution.
218 citations
TL;DR: In this article, the homotopy perturbation method (HPM) proposed in 1998 is only a special case of the HAM method, and it is shown that the solutions given by HPM (Siddiqui, A.M., Ahmed, M., Ghori, Q.K.).
Abstract: In this paper, we prove in general that the homotopy perturbation method (HPM) proposed in 1998 is only a special case of the homotopy analysis method (HAM) profound in 1992 when ħ = −1. Besides, by using the thin film flows of Sisko and Oldroyd 6-constant fluids on a moving belt as examples, we show that the solutions given by HPM (Siddiqui, A.M., Ahmed, M., Ghori, Q.K.: Chaos Solitons and Fractals (2006) in press) are divergent, and thus useless. However, by choosing a proper value of the auxiliary parameter ħ, we give convergent series solution by means of the HAM. These two examples also show that, different from the HPM and other traditional analytic techniques, the HAM indeed provides us with a simple way to ensure the convergence of the solution.
209 citations
TL;DR: In this paper, a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PIλDμ controllers is presented, which is based on plotting the global stability region in the (kp, ki)-plane for the PIλ controller and in the kp, kd)-space for the Dμ controller.
Abstract: This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PIλDμ controllers. It is based on plotting the global stability region in the (kp, ki)-plane for the PIλ controller and in the (kp, ki, kd)-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.
186 citations
TL;DR: In this article, a set of nonlinear energy sinks (NEs) are designed to be locally attached to a main structure, with the purpose of absorbing a significant part of the applied seismic energy, locally confining it and then dissipating it in the smallest possible time.
Abstract: In the field of seismic protection of structures, it is crucial to be able to diminish ‘as much as possible’ and dissipate ‘as fast as possible’ the load induced by seismic (vibration-shock) energy imparted to a structure by an earthquake. In this context, the concept of passive nonlinear energy pumping appears to be natural for application to seismic mitigation. Hence, the overall problem discussed in this paper can be formulated as follows: Design a set of nonlinear energy sinks (NESs) that are locally attached to a main structure, with the purpose of passively absorbing a significant part of the applied seismic energy, locally confining it and then dissipating it in the smallest possible time. Alternatively, the overall goal will be to demonstrate that it is feasible to passively divert the applied seismic energy from the main structure (to be protected) to a set of preferential nonlinear substructures (the set of NESs), where this energy is locally dissipated at a time scale fast enough to be of practical use for seismic mitigation. It is the aim of this work to show that the concept of nonlinear energy pumping is feasible for seismic mitigation. We consider a two degree-of-freedom (DOF) primary linear system (the structure to be protected) and study seismic-induced vibration control through the use of Vibro-Impact NESs (VI NESs). Also, we account for the possibility of attaching to the primary structure additional alternative NES configurations possessing essential but smooth nonlinearities (e.g., with no discontinuities). We study the performance of the NESs through a set of evaluation criteria. The damped nonlinear transitions that occur during the operation of the VI NESs are then studied by superimposing wavelet spectra of the nonlinear responses to appropriately defined frequency – energy plots (FEPs) of branches of periodic orbits of underlying Conservative systems.
182 citations
TL;DR: In this article, a series of two papers is devoted to detailed investigation of the response regimes of linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation.
Abstract: A series of two papers is devoted to detailed investigation of the response regimes of linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation. The first paper of the series is devoted to analytic and numeric description of the attractors (response regimes) of the system. Analytic approach is based on averaging and multiple-scales analysis, the mass ratio being used as the small parameter. The problem of possible coexistence of different attractors is reduced to analysis of flow on slow invariant manifolds (SIM) of the system. Numeric simulation confirms the predictions of the analytic model concerning the number, the shape, and the structure of the response regimes and reveals some other features of these attractors.
176 citations
TL;DR: In this paper, the performance of a strongly nonlinear, damped vibration absorber with relatively small mass attached to a periodically excited linear oscillator was investigated under harmonic external forcing and assessment of possible application of the energy sink for vibration absorption and mitigation.
Abstract: This paper is the second one in the series of two papers devoted to detailed investigation of the response regimes of a linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation. In this paper, we study the performance of a strongly nonlinear, damped vibration absorber with relatively small mass attached to a periodically excited linear oscillator. We present a nonlinear absorber tuning procedure in the vicinity of (1:1) resonance which provides the best total system energy suppression, using analytical and numerical tools. A linear absorber is also tuned according to the same criterion of total system energy suppression as the nonlinear one. Both optimally tuned absorbers are compared under common parameters of damping, external forcing but different absorber stiffness characteristics; certain cases for which nonlinear absorber is preferable over the linear one are revealed and confirmed numerically.
151 citations
TL;DR: In this article, the unscented Kalman filter (UKF) was proposed for softening single degree-of-freedom structural systems, and the performance of the UKF was shown to be significantly superior to that of the EKF in terms of state tracking and model calibration.
Abstract: Joint estimation of unknown model parameters and unobserved state components for stochastic, nonlinear dynamic systems is customarily pursued via the extended Kalman filter (EKF). However, in the presence of severe nonlinearities in the equations governing system evolution, the EKF can become unstable and accuracy of the estimates gets poor. To improve the results, in this paper we account for recent developments in the field of statistical linearization and propose an unscented Kalman filtering procedure. In the case of softening single degree-of-freedom structural systems, we show that the performance of the unscented Kalman filter (UKF), in terms of state tracking and model calibration, is significantly superior to that of the EKF.
136 citations
TL;DR: In this paper, the He's variational iteration method (VIM) is implemented to give approximate and analytical solutions for the Klein-Gordon equation, which is the relativistic version of the Schrodinger equation.
Abstract: In this paper, we present the solution of the Klein--Gordon equation. Klein--Gordon equation is the relativistic version of the Schrodinger equation, which is used to describe spinless particles. The He’s variational iteration method (VIM) is implemented to give approximate and analytical solutions for this equation. The variational iteration method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Application of variational iteration technique to this problem shows rapid convergence of the sequence constructed by this method to the exact solution. Moreover, this technique reduces the volume of calculations by avoiding discretization of the variables, linearization or small perturbations.
125 citations
TL;DR: In this article, the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface is considered and the homotopy analysis method is employed for the analytic solution.
Abstract: This study is concerned with the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface. Homotopy analysis method (HAM) is employed for the analytic solution. The similarity transformations have been used for reducing the partial differential equations into a system of two coupled ordinary differential equations. The series solution of the obtained system is developed and convergence of the results are explicitly given. The effects of the parameters M, s and λ on the velocity fields are presented graphically and discussed. It is worth mentioning here that for the shrinking surface the stable and convergent solutions are possible only for MHD flows.
TL;DR: In this paper, an analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables is presented.
Abstract: This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.
TL;DR: In this paper, a numerical scheme to solve the third-order nonlinear KdV equation using collocation points and approximating the solution using multiquadric (MQ) radial basis function (RBF) is presented.
Abstract: Recently, there has been an increasing interest in the study of initial boundary value problems for Korteweg–de Vries (KdV) equations. In this paper, we propose a numerical scheme to solve the third-order nonlinear KdV equation using collocation points and approximating the solution using multiquadric (MQ) radial basis function (RBF). The scheme works in a similar fashion as finite-difference methods. Numerical examples are given to confirm the good accuracy of the presented scheme.
TL;DR: In this article, the relationship between the parameters that appear in the differential equation and the shape of the obtained hysteresis loop is explored by analytically exploring this relationship using a new form of the model called the normalized one.
Abstract: The Bouc–Wen model for smooth hysteresis has received an increasing interest in the last few years due to the ease of its numerical implementation and its ability to represent a wide range of hysteresis loop shapes. This model consists of a first-order nonlinear differential equation that contains some parameters that can be chosen, using identification procedures, to approximate the behavior of given physical hysteretic system. Despite a large body of literature dedicated to the Bouc–Wen model, the relationship between the parameters that appear in the differential equation and the shape of the obtained hysteresis loop is little understood. The objective of this paper is to fill this gap by analytically exploring this relationship using a new form of the model called the normalized one. The mathematical framework introduced in this study formalizes the vague notion of “loop shape" into precise quantities whose variation with the Bouc–Wen model parameters is analyzed. In light of this analysis, the parameters of Bouc–Wen model are re-interpreted.
TL;DR: In this article, the nonlinear planar vibration of a pipe conveying pulsatile fluid subjected to principal parametric resonance in the presence of internal resonance is investigated using the method of multiple scales (MMS).
Abstract: In this paper, the nonlinear planar vibration of a pipe conveying pulsatile fluid subjected to principal parametric resonance in the presence of internal resonance is investigated. The pipe is hinged to two immovable supports at both ends and conveys fluid at a velocity with a harmonically varying component over a constant mean velocity. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the pipe. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of mean flow velocity, resulting in a three-to-one internal resonance. The analysis is done using the method of multiple scales (MMS) by directly attacking the governing nonlinear integral-partial-differential equations and the associated boundary conditions. The resulting set of first-order ordinary differential equations governing the modulation of amplitude and phase is analyzed numerically for principal parametric resonance of first mode. Stability, bifurcation, and response behavior of the pipe are investigated. The results show new zones of instability due to the presence of internal resonance. A wide array of dynamical behavior is observed, illustrating the influence of internal resonance.
TL;DR: In this article, a new absolute nodal coordinate-based finite element was introduced to solve the locking problem of a shear deformable element based on the NN formulation, where the position of any point of the element volume is defined employing independent slope coordinates.
Abstract: The absolute nodal coordinate formulation has been recently extended to shear deformable beam or plate elements. This has been accomplished, in practice, by parameterizing the complete volume of the elements instead of a line or surface in the element kinematics description. In the absolute nodal coordinate formulation, the position of any point of the element volume is defined employing independent slope coordinates. The use of a large number of slope coordinates leads to unusual kinematic features that must be accounted for in order to avoid the element locking. This study demonstrates that the shear deformable element based on the absolute nodal coordinate formulation suffers from curvature thickness locking and shear locking in addition to the previously reported Poisson’s locking. Due to the tendency of locking, the use of the absolute nodal coordinate formulation can lead to elements with weak performance. In order to eliminate locking problems, this study introduces a new absolute nodal coordinate-based finite element. The introduced element uses redefined polynomial expansion together with a reduced integration procedure. The performance of the introduced element is studied by means of certain dynamic problems. The element exhibits a competent convergence rate and it does not suffer from the previously mentioned locking effects.
TL;DR: In this article, a new tuning method for fractional PIα controllers is described, which takes advantage of the fractional order α to offer an excellent tradeoff between dynamic performances and stability robustness.
Abstract: This paper describes a new tuning method for fractional PIα controllers. The main theoretical contribution of the paper is the analytical solution of a nonlinear function minimization problem, which plays a central role in deriving the tuning formulae. These formulae take advantage of the fractional order α to offer an excellent tradeoff between dynamic performances and stability robustness. Finally, a position control is implemented to compare laboratory experiments with computer simulations. The comparison results show the good performance of the tuning formulae.
TL;DR: In this paper, the desired system outputs, expressed in terms of the system states, are treated as servo-constraints on the system, and the problem is viewed from the constrained motion perspective.
Abstract: This paper deals with a class of controlled mechanical systems in which the number of control inputs, equal to the number of desired system outputs, is smaller than the number of degrees of freedom. The related inverse dynamics control problem, i.e., the determination of control input strategy that force the underactuated system to complete the partly specified motion, is a challenging task. In the present formulation, the desired system outputs, expressed in terms of the system states, are treated as servo-constraints on the system, and the problem is viewed from the constrained motion perspective. Mixed orthogonal-tangent realization of the constraints by the available control reactions is stated, and a specialized methodology for solving the “singular” control problem is developed. The governing equations are manipulated to index three differential-algebraic equations, and a simple numerical code for solving the equations is proposed. The feedforward control law obtained as a solution to these equations can then be enhanced by a closed-loop control strategy with feedback of the actual servo-constraint violations to provide stable tracking of the reference motion in the presence of perturbations and modeling uncertainties. An overhead trolley crane executing a load-prescribed motion serves as an illustration. Some results of numerical simulations are reported.
TL;DR: In this article, the second-order multiple scales solution of a kinematically non-condensed cable model was used to investigate the effects of both non-linear dynamic extensibility and system asymmetry due to inclined sagged configurations.
Abstract: Resonant multi-modal dynamics due to planar 2:1 internal resonances in the non-linear, finite-amplitude, free vibrations of horizontal/inclined cables are parametrically investigated based on the second-order multiple scales solution in Part I [1] (in press). The already validated kinematically non-condensed cable model accounts for the effects of both non-linear dynamic extensibility and system asymmetry due to inclined sagged configurations. Actual activation of 2:1 resonances is discussed, enlightening on a remarkable qualitative difference of horizontal/inclined cables as regards non-linear orthogonality properties of normal modes. Based on the analysis of modal contribution and solution convergence of various resonant cables, hints are obtained on proper reduced-order model selections from the asymptotic solution accounting for higher-order effects of quadratic nonlinearities. The dependence of resonant dynamics on coupled vibration amplitudes, and the significant effects of cable sag, inclination and extensibility on system non-linear behavior are highlighted, along with meaningful contributions of longitudinal dynamics. The spatio-temporal variation of non-linear dynamic configurations and dynamic tensions associated with 2:1 resonant non-linear normal modes is illustrated. Overall, the analytical predictions are validated by finite difference-based numerical investigations of the original partial-differential equations of motion.
TL;DR: The authors used agent-based neural network models (Neugents) to analyse the impacts of socioeconomic, context and information variables on individual behaviour and propensity to change route and adjust travel patterns.
Abstract: This paper addresses commuters' route choice behaviour in response to traveller information systems. The data used in this study was obtained from a field behavioural survey of drivers that was conducted on a congested commuting corridor in Brisbane, Australia. Agent-based neural network models (Neugents) were used to analyse the impacts of socio-economic, context and information variables on individual behaviour and propensity to change route and adjust travel patterns. The results from these models clearly indicate that prescriptive, predictive and quantitative real-time delay information provided for both the usual and best alternate routes are most effective in influencing commuters to change their routes. The Neugent behavioural models describing drivers' dynamic route choice decision making were also implemented within a microscopic traffic simulation tool to evaluate the corridor-wide impacts of providing drivers with real-time traffic information. The simulation results support the notions that commuters' decisions to divert to alternate routes are influenced by their socio-economic characteristics; the degree of familiarity with network conditions and the expectation of an improvement in travel time that exceeds a certain delay threshold associated with each commuter. An evaluation of the benefits of the Neugent model over static route choice algorithms which do not consider dynamic driver behaviour and compliance with travel advice showed improvements of 4–7% in network speeds; 5–8% in network delays; 7–11% in stop time per vehicle and 1–3% in network travel times.
TL;DR: In this article, a chaotic complex nonlinear system is introduced and its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos.
Abstract: In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.
TL;DR: Recently, geometric singular perturbation theory has been extended considerably while at the same time producing many new applications as mentioned in this paper, and a number of aspects relevant to non-linear dynamics to apply this to periodic solutions within slow manifolds.
Abstract: Recently, geometric singular perturbation theory has been extended considerably while at the same time producing many new applications. We will review a number of aspects relevant to non-linear dynamics to apply this to periodic solutions within slow manifolds and to review a number of non-hyperbolic cases. The results are illustrated by examples.
TL;DR: In this article, two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation.
Abstract: In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.
TL;DR: In this article, the stability of a milling process is studied by using a semi-discretization method, which includes loss-of-contact effects between the workpiece and the tool and time-delay effects associated with the chip-thickness variation.
Abstract: In this article, the stability of a milling process is studied by using a semi-discretization method. The model of the workpiece–tool system includes loss-of-contact effects between the workpiece and the tool and time-delay effects associated with the chip-thickness variation. In addition, feed-rate effects are also considered. The governing system of equations is a non-autonomous, delay-differential system with time-periodic coefficients. Stability of periodic orbits of this system is studied to predict the onset of chatter and numerical evidence is provided for period-doubling bifurcations and secondary Hopf bifurcations. Stability charts generated using the semi-discretization method are found to compare well with the corresponding results obtained through time-domain simulations.
TL;DR: In this paper, recursive matrix relations for kinematics and dynamics of the HALF parallel manipulator with revolute actuators are presented, assuming that the position and the motion of the moving platform are known.
Abstract: Recursive matrix relations for kinematics and dynamics of the HALF parallel manipulator are presented in this paper. The prototype of this robot is a spatial mechanism with revolute actuators, which has two translation degrees of freedom and one rotation degree of freedom. The parallel manipulator consists of a base plate, a movable platform and a system of three connecting legs, having wide application in the fields of industrial robots, simulators, parallel machine tools and any other manipulating devices where high mobility is required. Supposing that the position and the motion of the moving platform are known, an inverse dynamics problem is solved using the principle of virtual powers. Finally, some iterative matrix relations and graphs of the torques and powers for all actuators are analysed and determined. It is shown that this approach is an effective means for kinematics and dynamics modelling of parallel mechanisms.
TL;DR: In this paper, a time-delay model for prey-predator growth with stage-structure is considered, and the stability and Hopf bifurcations are investigated by analyzing the distribution of the roots of associated characteristic equation.
Abstract: A time-delay model for prey–predator growth with stage-structure is considered. At first, we investigate the stability and Hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation. Then, an explicit formula for determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations is derived, using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out for supporting the analytic results.
TL;DR: In this article, a two-dimensional microslip friction model with normal load variation induced by normal motion is presented, which characterizes the stick-slip-separation of the contact interface and determines the resulting friction force, including its time variance and spatial distribution, between two elastic structures.
Abstract: A two-dimensional microslip friction model with normal load variation induced by normal motion is presented in this paper. The model is a distributed parameter model, which characterizes the stick-slip-separation of the contact interface and determines the resulting friction force, including its time variance and spatial distribution, between two elastic structures. When the relative motion is simple harmonic motion, the stick-slip-separation transition angles associated with any point in the contact area can be analytically determined within a cycle of motion. In addition, if the relative motion is given, stick-slip-separation transition boundaries inside the contact area and their time variances can be determined. Along with an iterative multi-mode solution approach utilizing harmonic balance method (HBM), the developed model can be employed to determine the forced response of frictionally constrained structures. In the approach, the forced response is constructed in terms of the free mode shapes of the structure; consequently, it can be determined at any excitation frequency and for any type of normal load distribution. Two examples, a one-dimensional beam like damper and a more realistic blade to ground damper, are employed to illustrate the predictive abilities of the developed model. It is shown that while employing a single mode model, transition boundaries for the beam like damper agrees with the results given in the literature, the developed method identifies the phase difference along the slip to stick transition boundary when a multi-mode model is employed. Moreover, while partial slip is illustrated in the two examples, typical softening and hardening effects, due to separation of the contact surface, are also predicted for the blade to ground damper.
TL;DR: In this paper, the authors deal with nonlinear energy pumping which consists in passive irreversible transfer of energy from a linear structure to a nonlinear one, and various results about energy pumping based on recent works are given.
Abstract: The present study deals with nonlinear energy pumping which consists in passive irreversible transfer of energy from a linear structure to a nonlinear one. Various results (theoretical, numerical, and experimental) about energy pumping based on recent works are given. Thus, the phenomenon is studied for different excitations: transient and periodical. Moreover, advantages of such a system are carried out in particular efficiency of this phenomenon. That is why the robustness and comparison with classical tuned mass damper are analyzed. An application is considered with physical experiment using a reduced scale building.
TL;DR: In this article, a radial cam and a flat-faced follower are used for the analysis of piecewise-smooth dynamical systems with impacts, where the follower is observed to detach from the cam and then show the emergence of periodic impacting behavior characterized by many impacts and chattering.
Abstract: In this paper, we present the design, modelling and experimental validation of a novel experimental cam-follower rig for the analysis of bifurcations and chaos in piecewise-smooth dynamical systems with impacts. Experimental results are presented for a cam-follower system characterized by a radial cam and a flat-faced follower. Under variation of the cam rotational speed, the follower is observed to detach from the cam and then show the emergence of periodic impacting behaviour characterized by many impacts and chattering. Further variations of the cam speed cause the sudden transition to seemingly aperiodic behaviour. These results are compared with the numerical simulation of a mathematical model of the system which shows the same qualitative behaviour. Excellent quantitative agreement is found between the numerical and experimental results.
TL;DR: In this paper, a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities is studied, and it is shown that the attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES).
Abstract: We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency–energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers.