Resonance Case Research Articles

Internal, primary and combination resonances of a wind turbine blade with coupled flapwise and edgewise motions

In recent decades blades of wind turbines have been designed longer and longer for better performance of energy harvesting. The growth in structural flexibility has strengthened the effect of geometric nonlinearity and led to complicated nonlinear dynamic behavior of vibration response. In this paper, an investigation on internal, primary and multi-mode combination resonances of a flexible wind turbine blade with coupled flapwise and edgewise motions is studied. Through the Hamilton's principle, the dynamical model is established using the Euler-Bernoulli beam considering the actions of gravitational and unsteady aerodynamic force. The displacement of the blade is expanded into four basic modes based on the Galerkin's method, three in flapwise direction and one in edgewise direction. The governing equations of the four modal coordinates are solved using the method of multiple scales in transient and steady-state. For the transient responses, the internal and multi-mode combination resonances are revealed by the transfer of energy between resonant modes. Unlike the case of primary and internal resonances, the quasi-periodic under the multi-mode combination resonance is discovered through phase diagram of the first edgewise mode. For the steady-state responses, amplitude-frequency curves and phase diagrams of resonance modes under the primary and multi-mode combination resonance are derived. Influence of the combination resonance on the primary resonance is discussed for the steady-state responses with different aerodynamic force and geometric nonlinearity. Finally, parametric analyses are presented to show how steady-state motion of flapwise and edgewise modes vary with detuning parameters and design parameters.

Nonlinear Stability of Equilibrium Points in the Planar Equilateral Restricted Mass-Unequal Four-Body Problem

In this paper, we investigate the stability of equilibrium points for the planar restricted equilateral four-body problem in the case that one particle of negligible mass is moving under the Newtonian gravitational attraction of three positive masses [Formula: see text], [Formula: see text] and [Formula: see text] (called primaries). These always lie at the vertices of an equilateral triangle (Lagrangian configuration) and move with constant angular velocity in circular orbits around their center of masses. We consider the case where all the primaries have unequal masses, and investigate the nonlinear stability (in the sense of Lyapunov) of the elliptic equilibrium for the specific values of the mass [Formula: see text] and [Formula: see text] of the primary, fixed on the horizontal axis. Moreover, the [Formula: see text][Formula: see text]:[Formula: see text][Formula: see text] four-order resonant cases are determined and the stability is investigated. In this study, Markeev’s theorem and Arnold’s theorem become key ingredients.

Flavour mixing transport theory and resonant leptogenesis

We derive non-equilibrium quantum transport equations for flavour-mixing fermions. We develop the formalism mostly in the context of resonant leptogenesis with two mixing Majorana fermions and one lepton flavour, but our master equations are valid more generally in homogeneous and isotropic systems. We give a hierarchy of quantum kinetic equations, valid at different approximations, that can accommodate helicity and arbitrary mass differences. In the mass-degenerate limit the equations take the familiar form of density matrix equations. We also derive the semiclassical Boltzmann limit of our equations, including the CP-violating source, whose regulator corresponds to the flavour coherence damping rate. Boltzmann equations are accurate and insensitive to the particular form of the regulator in the weakly resonant case ∆m » Γ, but for ∆m ≲ Γ they are qualitatively correct at best, and their accuracy crucially depends on the form of the CP-violating source.

On the motion of a triple pendulum system under the influence of excitation force and torque

In this article, a nonlinear dynamical system with three degrees of freedom (DOF) consisting of multiple pendulums (MP) is investigated. The motion of this system is restricted to be in a vertical plane, in which its pivot point moves in a circular path with constant angular velocity, under the action of an external harmonic force and a moment acting perpendicular to the direction of the last arm of MP and at the suspension point respectively. Multiple scales technique (MST) among other perturbation methods is used to obtain the approximate solutions of the equations of motion up to the third approximation because it is authorizing to execute a specific analysis of the system behaviour and to realize the solvability conditions given the resonance cases. The stability of the considered dynamical model utilizing the nonlinear stability analysis approach is examined. The solutions diagrams and resonance curves are drawn to illustrate the extent of the effect of various parameters on the solutions. The importance of this work is due to its uses in human or robotic walking analysis.

Exploiting internal resonance to improve flow energy harvesting from vortex-induced vibrations

Vortex-induced vibrations have been recently employed to capture scalable flow energy harvesters, which can attain the maximum power when the wind speed is in the lock-in region where the vortex-shedding frequency is close to the structural frequency. Nevertheless, the dynamical mechanism of the synchronization phenomenon has not been reported. To solve this critical problem, in this paper we explore a novel internal resonance to scavenge flow energy from vortex-induced vibrations, the mechanism of synchronous oscillations is introduced by the amplitude-frequency relationship and confirmed by the energy function. To show the capturing performance, an electromagnetic energy harvester with one-to-one internal resonance is proposed. Based on the harmonic balance method, the electromechanical coupling equations are decoupled, and the first order approximate harmonic responses of displacement and current are established. The modulation equations are derived, the amplitude-frequency curves of displacement and current are plotted with different detuning parameters. The advantage of the proposed one-to-one internal resonance is compared to the noninternal resonance case, the results express that the internal resonance scheme can enhance flow energy capture. The effects of physical parameters on the scavenged power are discussed. The accuracy and efficiency of the approximate analytical results are checked by numerical simulations.

Study on Linear and Nonlinear Dynamically P-delta Effects on Frame Structures under earthquake spectra, harmonic and stochastic excitations

In this paper the effect of axial load on dynamic behavior of a simple frame, subjected to harmonic, seismic and earthquake excitation is investigated. The equations of motion are considered for two types of small and large deformations. The method of multiple scales is applied to solve the differential equations of motion with harmonic loading and for small and large deformations. Then, the steady state response near one-to-one resonance condition is studied. The results show that the dynamic behavior of the frame under axial load is completely different in resonance and non-resonance cases. The equations of motion with earthquake loading is also considered and the effect of axial load in the frame behavior under the time history and the response spectrum of the model is studied. Although white noise as a stochastic loading is applied to the model and, the results are approximated using the method of stochastic differential equations so, the mean value and covariance are calculated and the effect of axial force on them is investigated.

Nonlinear Dynamics of Suspended Cables under Periodic Excitation in Thermal Environments: Two-to-One Internal Resonance

The temperature change is a non-negligible factor in examining the vibration behaviors of the cable structures. For this reason, the paper aims at investigating the thermal effects on suspended cables’ resonant responses considering two-to-one internal resonances. Firstly, a nonlinear continuous condensed model of the suspended cable under periodic excitation in thermal environments is adopted. Then, a multidimensional discretized model is constructed via the Galerkin method. Following the multiple scaling procedure, the modulation equations with both polar and Cartesian forms are obtained, which are solved numerically. A complete dynamic scenario is presented through bifurcation diagrams, phase portraits, time history curves, Fourier spectra, and Poincaré sections in three internal resonant cases. Numerical examples show that a small change in the static configuration due to thermal effects induces some noticeable changes in dynamic behaviors. The response amplitude, the nonlinear spring behavior, the resonant and stability region, the multi-periodic and chaotic motions are all dependent on temperature changes. Additional Hopf bifurcations might be found due to temperature changes, and it may lead to some more complicated dynamic characteristics. A good agreement between the perturbation and numerical solutions is observed to confirm the results’ correctness and accuracy.

The stability analysis for the motion of a nonlinear damped vibrating dynamical system with three-degrees-of-freedom

This article investigates the motion of double nonlinear damped spring pendulums with 3DOF in which its pivot point moves in an elliptic trajectory. The controlling system of motion is derived applying the Lagrange's equations and solved asymptotically up to the third approximation utilizing the multiple scales technique. The arising resonance cases are categorized in view of the solvability conditions in which the equations of modulation are obtained. The steady-state solutions in accordance with the stability and instability criteria are examined. The novelty of this work lies in obtaining high accuracy solutions compared with the previous works and in studying of the nonlinear stability analysis to identify the stability and instability areas. The time histories of the attained results, the response resonance curves, and the regions of stability are discussed and represented graphically to explore the good influence of the distinct parameters on the dynamical behaviour of the investigated system. The importance of this work is due to its great uses in various engineering applications.

Capillary‐gravity solitary waves on water of finite depth interacting with a linear shear current

AbstractThe problem of two‐dimensional capillary‐gravity waves on an inviscid fluid of finite depth interacting with a linear shear current is considered. The shear current breaks the symmetry of the irrotational problem and supports simultaneously counter‐propagating waves of different types: Korteweg de‐Vries (KdV)‐type long solitary waves and wave‐packet solitary waves whose envelopes are associated with the nonlinear Schrödinger equation. A simple intuition for the broken symmetry is that the current modifies the Bond number differently for left‐ and right‐propagating waves. Weakly nonlinear theories are developed in general and for two particular resonant cases: the case of second harmonic resonance and long‐wave/short‐wave interaction. Traveling‐wave solutions and their dynamics in the full Euler equations are computed numerically using a time‐dependent conformal mapping technique, and compared to some weakly nonlinear solutions. Additional attention is paid to branches of elevation generalized solitary waves of KdV type: although true embedded solitary waves are not detected on these branches, it is found that periodic wavetrains on their tails can be arbitrarily small as the vorticity increases. Excitation of waves by moving pressure distributions and modulational instabilities of the periodic waves in the resonant cases described above are also examined by the fully nonlinear computations.

Effects of geometrical nonlinearity on the performance of bidirectional tuned mass dampers

AbstractThis research work focuses on the presentation of a novel model for considering the effects of geometrical nonlinearity in the response of bidirectional tuned mass dampers (TMDs). A new form of nonlinear equations of motion is derived. Unlike the conventional linear model, herein, the equations of motion are coupled in two orthogonal directions. The response of the novel nonlinear equations is obtained and compared to the typical linear equations and it is shown that the linear ones are not accurate enough to model the response of the multidirectional TMDs. Perturbation analysis is done to assess the response of TMD system under two internal resonance cases. In the proposed model, phenomena such as jump and energy transfer can take place. The effects of excitation amplitude, damping ratios, and frequency detuning parameters are investigated on the jump and energy transfer phenomena. According to the proposed nonlinear equations, the performance of the bidirectional TMDs depends on the excitation amplitude. Moreover, in some cases, by considering the nonlinear terms in the bidirectional motion equations of TMDs, their performance is improved. However, the nonlinear terms may affect the performance of TMDs negatively, which leads to a very high amplitude of the responses in the main controlled structure under seismic and harmonic excitations.

New controller (NPDCVF) outcome of FG cylindrical shell structure

This paper presents a new active controller technique effect of functionally graded (FG) cylindrical shell structure model within joint harmonic and parametric excitations. The new active controller involves Nonlinear Proportional-Derivative (NPD) plus Negative Cubic Velocity Feedback (NCVF) as a new nonlinear control method (NPDCVF). The motivation of this study is the well-known observation that NPDCVF control method can offer a means to improve the performance of plant systems. The coupled nonlinear differential equations with two modes have been derived applying the von Kármán nonlinear theory, Galerkin’s process, and the static condensation technique. Static condensation is the process of decreasing the number of free displacements or degrees of freedom. We have added other three different controller methods over the structure to select the best controller. The three applied controller methods to the considered structure are: Integral Resonant Control (IRC), Positive Position Feedback (PPF), and Nonlinear Integral Positive Position Feedback (NIPPF) which are applied to the considered structure. We establish that the best one for dipping the high vibration amplitudes is the NPDCVF as a new controller. The perturbation technique is useful to solve the present model including quadratic and cubic nonlinearities subjected to mixed harmonic and parametric forces within the simultaneous primary resonance case (Ω=ω1,Ω=ω2) as the worst resonance case of the system. All numerical outcomes have been completed with the help of MATLAB Ra 18.0 program software over the investigated model. The obtained results have proved that this new controller is excellent for improving the structure by reducing the risky vibrations caused during primary resonances. Frequency response equations have helped in observing the stability examination of the obtained numerical solutions. The actions of several factors performed on the controlled model have been examined and described. Some comparisons have also been prepared with the available recent published papers of a similar model.

The dynamical analysis for the motion of a harmonically two degrees of freedom damped spring pendulum in an elliptic trajectory

This article investigates the motion of two degrees of freedom non-linear dynamical system represented by a spring pendulum in which its suspension point moves in an elliptic trajectory. Lagrange's equations are utilized to derive the governing equations of motion in light of the generalized coordinates of this system. The asymptotic solutions for the governing equations of motion up to the third approximation are obtained using the multiple scales technique. The different resonance cases are classified according to the modulation equations. The steady-state solutions are verified in the presence of the solvability conditions. The time histories of the considered motion, the resonance curves, and steady-state solutions are presented graphically. In addition to the simulations of the non-linear equations of the system's evolution, the stability criteria are carried out. The stability and instability regions are investigated, in which it is found the behaviour of the system is stable for a large number of the system’s parameters. In various engineering applications, a better understanding of the vibrational motion near resonance is critical, in which it can be avoided continued exposure to events that can cause severe damage.

On stability of a collinear libration point in the planar circular restricted photogravitational three-body problem in the cases of first and second order resonances

We deal with the planar circular photogravitational three-body problem. That is, we consider the motion of a particle under influence of gravitational and radiation forces acting from two bodies, which move in circular orbits. The stability of collinear point L1 , which located on the line between the bodies, is investigated. By using the method of normal forms and applying theorems of KAM theory we perform a nonlinear stability study for parameter values corresponding to the cases of first and second order resonances. Rigorous conclusions on instability and stability in the sense of Lyapunov have been obtained.

Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

The current study examines the hybrid Rayleigh–Van der Pol–Duffing oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincaré–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near fixed points. The phase portraits are plotted to visualize the behavior of HRVD around their fixed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisfied by making a comparison among them.

Morse inequalities at infinity for a resonant mean field equation

In this paper, we study the following mean field type equation: [Formula: see text] where [Formula: see text] is a closed oriented surface of unit volume [Formula: see text] = 1, [Formula: see text] positive smooth function and [Formula: see text], [Formula: see text]. Building on the critical points at infinity approach initiated in [M. Ahmedou, M. Ben Ayed and M. Lucia, On a resonant mean field type equation: A “critical point at infinity” approach, Discrete Contin. Dyn. Syst. 37(4) (2017) 1789–1818] we develop, under generic condition on the function K and the metric g, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters [Formula: see text]. We derive from these Morse inequalities at infinity various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. [Formula: see text].

Vibration Control for a Coupled Pitch- roll Ship Model Via a Negative Cubic Velocity Feedback Control

One of the most essential ship reactions to waves is roll motion. Due to the intricacy of ship wave interactions and their sensitivity, predicting such a reaction is extremely challenging. Because vibration motion is an undesirable occurrence, it must be removed, decreased, or controlled. A coupled Pitch- roll ship model with negative cubic velocity feedback control subjected to parametric excitations is premeditated and solved in this paper. The method of multiple time scales is applied to scrutinize the response of the two modes of the system neighbouring the simultaneous sub-harmonic, and internal resonance situation. Besides, the steady-state solution is determined through the Rung-Kutta Method (RKM) of fourth order. Stability of the steady state solution near this resonance case is discussed and studied applying Lyapunov’s first indirect method and Routh- Hurwitz criterion. The influences of the different parameters on the steady state solution are reconnoitred and discussed. The controller effects on the stability are clarified. Simulation results are accomplished with the help of MATLAB and Maple software programs.

Controllable probe absorption spectrum via vortex beams excitation in a cascaded atomic system

Dynamic control of probe absorption spectrum via vortex beams in multilevel systems has attracted much attention because it provides an avenue to observe unique transport properties. In this paper, we employ experimentally accessible parameters to flexibly control the probe absorption spectrum that shows vortex-induced transparency (VIT) and absorption (VIA) phenomena. Specifically, this system can be served as an optical switch while the vortex beams can act as the control knob, which has the ability to switch from a transparency dip to an absorption peak. Differing from a conventional electromagnetically induced transparency technique, our proposed scheme investigates VIT and VIA by tuning the orbital angular momentum of vortex beams in the resonance and non-resonance cases. Moreover, the collective feature caused by constructive or destructive interference is also studied via adjustable frequency detuning. Our results may provide an efficient approach to study similar phenomena in multilevel quantum systems driven by optical vortex beams.

Random Response of Wake-Oscillator Excited by Fluctuating Wind

Abstract Many wake-oscillator models applied to study vortex-induced vibration (VIV) are assumed to be excited by ideal wind that is assumed to be uniform flow with constant velocity. However, in the field of wind engineering, the real wind is generally described as being composed of mean wind and fluctuating wind. A wake-oscillator excited by fluctuating wind should be treated as a randomly excited and dissipated multi-degree-of-freedom (DOF) nonlinear system. The study of such a system is challenging and so far there is no exact solution available. The present paper aims to carry out the study on the stochastic dynamics of VIV. The stochastic averaging method of quasi-integrable Hamiltonian systems under wideband random excitation is applied to study both the original Hartlen–Currie wake-oscillator model and a modified version excited by fluctuating wind. The probability and statistics of the random response of wake-oscillator in resonant (lock-in) case and in non-resonant case are analytically obtained, and the analytical results are confirmed using numerical simulation of the original system.

Harmonic solutions for a class of non-autonomous piecewise linear oscillators

The study deals with the existence and uniqueness conditions of harmonic solutions of a class of piecewise linear oscillators with a general periodic excitation. Based on the relationship between natural frequency and driving frequency, we divide the discussion of the harmonic solutions into resonant case and non-resonant case. For resonant case, the existence conditions dependent on periodic excitation and clearance of harmonic solutions are given based on the Poincaré-Bohl fixed point theorem. Moreover, we give a necessary and sufficient condition of bounded solutions, where harmonic solutions, n-subharmonic solutions, and quasi-periodic solutions can occur simultaneously. For non-resonant case, the existence of harmonic solutions are analyzed based on contract mapping. The uniqueness of harmonic solutions for resonant and non-resonant cases are proven by variational method and reduction to absurdity. Numerical examples are presented to illustrate the theoretical results.

Flexural-Torsional Galloping of Prismatic Structures with Double-Symmetric Cross-Section

The linear galloping of prismatic structures having double-symmetric cross-section, subjected to steady wind flow acting along a symmetry axis, is investigated. The continuous system is reduced to a three degree-of-freedom system via a Galerkin approach. The quasi-steady assumption for the aerodynamic forces is applied, under the hypothesis that the galloping instability is well-separated from the vortex induced vibration phenomenon. Due to the structural symmetry conditions and accounting for the aerodynamic coupling, galloping is of flexural-torsional type, occurring in the direction orthogonal to the incident wind. Moreover, coupling is stronger close to the resonance between the flexural and torsional degrees of freedom. A linear stability diagram is built up in a two-parameter space, highlighting the role of coupling in modifying the critical wind velocity, and in producing a veering phenomenon between the two modes. The existence of points at which a double-Hopf bifurcation manifests itself is detected. Both exact and perturbation solutions are provided, these latter in the non-resonant and resonant cases, useful to throw light on the interactive mechanisms. The resonant perturbation solution permits to analytically investigate under which conditions coupling has a detrimental effect on galloping, which manifests at a wind velocity lower than the flexural and torsional critical velocities. Situations where coupling between modes leads to beneficial effect with respect to the Den Hartog's critical wind velocity are also highlighted. As an application, galloping of a family of multi-story tower buildings having a square cross-section is studied.