External Harmonic Force Research Articles

Dynamics of a time delayed Duffing oscillator

This paper focuses on the primary resonance of the time delayed Duffing oscillator solved by means of the multiple scales method. The second order approximation is used because the first approximation applied in the literature does not exhibit specific phenomena of the studied system. The Duffing system is investigated in two aspects. First of all, how delay displacement feedback influences the primary resonance of the classical Duffing oscillator in order to control the system and, on the other hand, how the external harmonic force influences vibrations of the system with time delay which is essential in applications, e.g. in machining. Stability of solutions and their bifurcations caused by the system parameters are shown. Especially, coexistence of possible stable and unstable solutions is investigated versus time delay and external excitation parameters. The selected analytical results are compared with numerical simulations. The model presented in the paper is considered as a general problem of non-linear oscillator with time delay but specific attention is paid to cutting process control.

Stochastic resonance in a nonlinear system with a 1/f spectrum

The system of two nonlinear stochastic equations simulating 1/f fluctuations during the interaction of nonequilibrium phase transitions in the presence of an external harmonic force is analyzed using numerical methods. It is shown that the stochastic resonance occurring in the system enhances the output periodic signal under the action of noise. A random process with a 1/f power spectrum corresponds to the Gibbs-Shannon information entropy peak. In stochastic resonance, the information entropy is minimal.

Analysis of Fluid Systems: Stability, Receptivity, Sensitivity

This article presents techniques for the analysis of fluid systems. It adopts an optimization-based point of view, formulating common concepts such as stability and receptivity in terms of a cost functional to be optimized subject to constraints given by the governing equations. This approach differs significantly from eigenvalue-based methods that cover the time-asymptotic limit for stability problems or the resonant limit for receptivity problems. Formal substitution of the solution operator for linear time-invariant systems results in the matrix exponential norm and the resolvent norm as measures to assess the optimal response to initial conditions or external harmonic forcing. The optimization-based approach can be extended by introducing adjoint variables that enforce governing equations and constraints. This step allows the analysis of far more general fluid systems, such as time-varying and nonlinear flows, and the investigation of wavemaker regions, structural sensitivities, and passive control strategies.

Sticking and nonsticking orbits for a two-degree-of-freedom oscillator excited by dry friction and harmonic loading

We consider a system composed of two masses connected by linear springs. One of the masses is in contact with a rough surface. Friction force, with Coulomb’s characteristics, acts between the mass and the surface. Moreover, the mass is also subjected to a harmonic external force. Several periodic orbits are obtained in closed form. A first kind of orbits involves sticking phases: during these parts of the orbit, the mass in contact with the rough surface remains at rest for a finite time. Another kind of orbits includes one or more stops of the mass with zero duration. Normal and abnormal stops are obtained. Moreover, for some of these periodic solutions, we prove that symmetry in space and time occurs.

A linear complementarity problem formulation for periodic solutions to unilateral contact problems

Presented is an approach for finding periodic responses of structural systems subject to unilateral contact conditions. No other non-linear terms, e.g. large displacements or strains, hyper-elasticity, plasticity, etc. are considered. The excitation period due to various forcing conditions—from harmonic external or contact forcing due to a moving contact interface—is discretized in time, such that the quantities of interest—displacement, velocity, acceleration as well as contact force—can be approximated through time-domain schemes such as backward difference, Galerkin, and Fourier. The solution is assumed to exist and is defined on a circle with circumference T to directly enforce its periodicity. The strategy for approximating time derivative terms within the discretized period, i.e. velocity and acceleration, is hence circulant in nature. This results in a global circulant algebraic system of equations with inequalities that can be translated into a unique Linear Complementarity Problem (LCP). The LCP is then solved by dedicated and established methods such as Lemke׳s Algorithm. This allows for the computation of approximate periodic solutions exactly satisfying unilateral contact constraints on a discrete time set. The implementation efficiency and accuracy are discussed in comparison to classical time marching techniques for initial value problems combined with a Lagrange Multiplier contact treatment. The LCP algorithm is validated using a simple academic problem. The extension to large-scale systems is made possible through the implementation of a Craig–Bampton type Modal Component Synthesis. The method shows applicability to industrial rotor/casing contact set-ups as shown by studying a compressor blade. A good agreement to time marching simulations is found, suggesting a viable alternative to time marching or Fourier-based harmonic balance simulations.

Nonlinear and quasi-linear behavior of a curved carbon nanotube vibrating in an electric force field; an analytical approach

The nonlinear vibrational model of a slightly curved single-walled carbon nanotube (SWCNT) resting on a Winkler-type elastic foundation is developed using nonlocal Euler- Bernoulli elastic theory. The SWCNT is assumed to vibrate under an external harmonic electric force field and an analytical solution is proposed to obtain the nonlinear resonant frequencies. The results show good agreement with the numerical simulation and the obtained analytical frequency is completely related to the curvature of the nanotube. Our model predicts that although the model is nonlinear in nature, the curved SWCNT could behave linearly in a certain amount of curvatures and this quasi-linear vibrational behavior of curved SWCNT is a function of aspect ratio, nonlocal parameter, and stiffness of the foundation.

Nonlinear stability and bifurcations of an axially moving beam in thermal environment

The thermo-mechanical nonlinear dynamics of an axially moving beam with coupled longitudinal and transverse displacements subjected to a distributed harmonic external force is numerically investigated. This includes a case where the system is tuned to a three-to-one internal resonance between the first two transverse modes and a case where it is not is considered. Two coupled nonlinear partial differential equations for the longitudinal and transverse motions are obtained using Hamilton’s principle and constitutive relations, as well as taking into account the thermal effects. The Galerkin method is then used to discretize these equations into a set of coupled nonlinear ordinary differential equations. Two different techniques are employed to solve the resulting equations; the pseudo-arclength continuation method and direct time integration to investigate the periodic vibrations and the global dynamics of the system, respectively. The effect of different parameters on the dynamics of the system is investigated through the frequency-response curves of the system and the bifurcation diagrams of Poincaré maps. Furthermore, time histories, phase-plane portraits, and fast Fourier transforms are presented for a few different system parameter sets. It is illustrated that the system shows a broad variety of rich dynamics, depending on system parameters and the temperature rise.

Hopf bifurcation and multistability in a system of phase oscillators

We study the phase reduction of two coupled van der Pol oscillators with asymmetric repulsive coupling under an external harmonic force. We show that the system of two phase oscillators undergoes a Hopf bifurcation and possesses multistability on a 2π-periodic phase plane. We describe the bifurcation mechanisms of formation of multistability in the phase-reduced system and show that the Andronov-Hopf bifurcation in the phase-reduced system is not an artifact of the reduction approach but, indeed, has its prototype in the nonreduced system. The bifurcational mechanisms presented in the paper enable one to describe synchronization effects in a wide class of interacting systems with repulsive coupling e.g., genetic oscillators.

ESTIMATION OF CHAOTIC THRESHOLDS FOR THE RECENTLY PROPOSED ROTATING PENDULUM

In this paper, we investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.

Generation of Chaotic Microwave Pulses in Ferromagnetic Film Ring Oscillators Under External Influence

This study reports on the experimental results of chaotic microwave (MW) pulse generation in a ferromagnetic film ring oscillator under pulse-modulated (PM) and narrowband noise forces. Chaotic dynamics of the investigated self-oscillatory system is produced through three-wave parametric interactions of spin waves. It is shown, that periodical sequences of chaotic MW pulses are generated under a PM force and solitary chaotic MW pulses are generated under a noise force. Such pulses are formed through forced chaos synchronization. For description of forced chaos synchronization phenomenon, a ferromagnetic film ring oscillator model with nonlinear amplification and delay time is constructed. This model is based on the use of the Landau-Lifshitz equation, magnetostatic approximation and the Holstein-Primakoff transformation. Calculation results of chaos suppression under an external harmonic force are demonstrated.

Steady Collision State in One, Two or Three-Body Vibro-Impact Oscillator with External Force or Enforced Displacement

By adding impulse excitation and energy dispersion operations, we obtain steady states of three-body vibro-impact system with two small oscillators coupled to one large oscillator under wave algorithm. From the steady collision conditions of the three-body system, we deduce a trajectory and a function of external force or enforced displacement for steady collision conditions of two kinds of two-body or single-body vibro-impact oscillator. By using either function, mass of the vibro-impact oscillator steadily collies with arbitrary impact positions and velocities. Compared with these functions, steady collisions were not achieved with harmonic external forces or enforced displacements. These functions could be favorable for steady impact process.

Energy Absorption by a Single Abrikosov’s Vortex in NbTi and YBaCuO Superconductors

The calculations of the absorption energy and the Q-factor of vibration of a vortex line under the effect of an external uniform harmonic force have been presented. The line tension of the vortex line and its effective mass, Magnus force, pinning force, and viscosity force were included in the equation of the dynamics. Two peaks of absorption in accordance with two modes of the oscillation spectrum were observed. The first peak of absorption is a result of flux pinning, which coincides with the De Gennes–Matricon solution in some limits. The second (high-frequency) peak is a result of a novel mode related to the effective mass of a vortex line. Temperature dependencies of absorption peaks were calculated for the NbTi and YBaCuO superconductors for orientations H∥c and H⊥c. The resonant frequency of the resonance peaks of absorption decreases with an increase in temperature. Some other peculiarities of the oscillating dynamics of the vortex line were discussed.

Vibration analysis of rotating 3D beams by the p-version finite element method

Moderately nonlinear vibrations of 3D beams with rectangular cross section and that rotate about a fixed axis are investigated by the p-version finite element method. Two types of nonlinearity are taken into account: one comes from the nonlinear strain–displacement relation, the other appears because of the inertia forces due to the rotation of the beam. The beam may experience longitudinal deformations, torsion and non-planar bending. The beam model is based on Timoshenko's theory for bending and Saint-Venant's for torsion, i.e. it is assumed that the cross section rotates about the longitudinal axis as a rigid body but may deform in longitudinal direction due to warping; furthermore torsion is not constrained to be constant. All inertia forces due to the rotation are included in the inertia terms in the equation of motion, which is derived by the principle of virtual work. The influence of the speed of rotation on the bending linear modes of vibration is presented. Then, nonlinear forced vibrations of rotating beams are investigated in the time domain, using direct integration of the equation of motion and considering constant and non-constant speed of rotation. Impulsive type and harmonic external forces are considered.

Dynamic analysis of externally excited NES-controlled systems via a mixed Multiple Scale/Harmonic Balance algorithm

A general, nonlinear, multi-d.o.f. structure, excited by harmonic external force in 1:1 resonance with one of the modes of the system, is considered. The structure is attached to an essentially nonlinear oscillator, with small mass and damping (Nonlinear Energy Sink, NES). The scope of the NES is to passively control the amplitude of vibrations of the main structure. A mixed Multiple Scale/Harmonic Balance Method (MSHBM) is proposed to get the differential equations describing the slow- and fast-flow dynamics of the whole structure. The main advantage of the procedure is that no complexification-averaging is required, so that the analysis is reconducted in the framework of the classical perturbation techniques.

Nonlinear phenomena in mechanical system dynamics

The goal of the paper is to present selected, untypical, and intuitively unexpected phenomena from nonlinear mechanics. Particular attention is paid to the dynamics of self-, parametric and external excited systems. Interactions between these various vibration types lead mainly to quasi-periodic responses. However, in the selected domains of system parameters, the effect of frequency locking is observed. Furthermore, external harmonic force imposed on such a system produces a specific internal loop inside a resonance zone. As an example of nonlinear autoparametric systems, a structure (oscillator) with an attached pendulum is presented. The nonlinear terms introduced by pendulum motion cause instabilities in the resonance region. This instability transits the pendulum to rotation or chaotic motion. An application of nonlinear couplings for the reduction of unwanted vibrations is also studied. In order to reduce vibrations, the main structure is coupled to an electrical oscillator by a quadratic term. It has been shown that such a coupling leads to the amplitude saturation phenomenon which can then be used to design a nonlinear control strategy.

New limit cycles of dry friction oscillators under harmonic load

We consider a system composed of two masses connected by linear springs. One of the masses is in contact with a driving belt moving at a constant velocity. Friction force, with Coulomb’s characteristics, acts between the mass and the belt. Moreover, the mass is also subjected to a harmonic external force. Several periodic orbits including stick phases and slip phases are obtained. In particular, the existence of periodic orbits including a part where the mass in contact with the belt moves in the same direction at a higher speed than the belt itself is proved. Non-sticking orbits are also found for a non-moving belt. We prove that this kind of solution is symmetric in space and in time.

Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis

The nonlinear coupled longitudinal-transverse vibrations and stability of an axially moving beam, subjected to a distributed harmonic external force, which is supported by an intermediate spring, are investigated. A case of three-to-one internal resonance as well as that of non-resonance is considered. The equations of motion are obtained via Hamilton’s principle and discretized into a set of coupled nonlinear ordinary differential equations using Galerkin’s method. The resulting equations are solved via two different techniques: the pseudo-arclength continuation method and direct time integration. The frequency-response curves of the system and the bifurcation diagrams of Poincare maps are analyzed.

СИНТЕЗ СИСТЕМИ КЕРУВАННЯ ВІБРОАКУСТИЧНИМ ВИПРОМІНЮВАННЯМ ГНУЧКОЇ КОНСТРУКЦІЇ

Розглянуто синтезовану систему керування, яка здатна зменшувати сталу вібраційнувідповідь тонкостінної гнучкої конструкції в широкій смузі низьких частот. Показано, що воснові алгоритму шаблонного пошуку лежить припущення про наявність зв’язку міжсигналом, що утворюється системою керування та усталеною вібраційною відповіддюгнучкої конструкції. Ефективність запропонованої системи керування проілюстрованоекспериментально на прикладі затиснутої пластини.Ключові слова: вібрація, гнучка конструкція, система керування, шаблонний пошук,шум.

Effect of Amplification on Acoustic and Vibration Energy with Vibroacoustic Coupling

In order to apply acoustic energy to industrial fields, its amplification is essential and vibroacoustic coupling is one of best ways of amplifying acoustic energy. In the present study, we consider a cylindrical structure with thin plate at both ends and investigate vibroacousitc coupling between the plate vibrations and internal sound field when an external harmonic force is applied to one end plate, regarding such a coupling phenomena as one of the available measures to amplify acoustic energy. This coupling is theoretically and experimentally investigated by considering the behavior of both plates and the acoustic characteristics of the internal sound field for varying dimensions of the end plates and the cylinder. In the analytical model, the end plates are supported circumferentially by springs in order to obtain support conditions that are as close as possible to the actual condition in the experiment due to adjustments of spring stiffnesses. The cylinder length and phase difference, which maximized the sound pressure level inside the cavity, are clarified theoretically. These theoretically results are verified experimentally through an excitation experiment using an experimental apparatus that is emulated by the analytical model. The above-describe results reveal such a vibroacoustic coupling phenomenon is useful for the amplification of the acoustic energy.

NONLINEAR RESPONSE OF THE MASS-SPRING MODEL WITH NONSMOOTH STIFFNESS

In this paper, we study the nonlinear response of the nonlinear mass-spring model with nonsmooth stiffness. For this purpose, we take as prototype model, a system that consists of the double-well smooth potential with an additional spring component acting on the system only for large enough displacement. We focus our study on the analysis of the homoclinic orbits for such nonlinear potential for which we observe the appearance of chaotic motion in the presence of damping effects and an external harmonic force, analyzing the crucial role of the linear spring in the dynamics of our system. The results are shown by using both the Melnikov analysis and numerical simulations. We expect our work to have implications on problems concerning the suspension of vehicles, among others.