External Harmonic Force Research Articles

Dynamic Analysis of Vibration Reduction and Energy Harvesting Using a Composite Cantilever Beam with Galfenol and a Nonlinear Energy Sink

This paper explores a new method for vibration control and energy harvesting by coupling a nonlinear energy sink (NES) with a cantilever beam containing giant magnetostrictive material. A mathematical model of the principal linear system under the action of an external harmonic force is established, and the dynamic equation of the system is derived using Newton’s second law, Hamilton’s principle, and nonlinear boundary conditions. The displacement responses of the system as functions of time and of frequency are obtained by using the Runge–Kutta method and harmonic balance method. The analytic results are found to be consistent with the numerical solution, confirming the accuracy and reliability of the calculated results. The frequency response and time response of the system with and without NES coupling, and with other parameters unchanged, were compared. The results show that the proposed structure can achieve excellent vibration suppression. The energy harvesting effect of the system as a function of time is also analyzed by the Runge–Kutta method, comparing results for different values of parameters such as NES mass, nonlinear spring stiffness, and damping coefficient. The results indicate that the proposed structure can achieve the desired effects of vibration reduction and energy harvesting, and that results can be optimized by adjusting the relevant parameters.

Time-squeezing and time-expanding transformations in harmonic force fields.

A variety of real life phenomena exhibit complex time-inhomogeneous nonlinear diffusive motion in the presence of an external harmonic force. In capturing the characteristics of such dynamics, the class of Ornstein-Uhlenbeck processes, with its physical time appropriately modulated, has long been regarded as the most relevant model on the basis of its mean reversion property. In this paper, we contrast two similar, yet definitely different, time-changing mechanisms in harmonic force fields by systematically deriving and presenting a variety of key properties all at once, that is, whether or not and how those time-changing mechanisms affect the characteristics of mean-reverting diffusion through sample path properties, the marginal probability density function, the asymptotic degeneracy of increments, the stationary law, the second-order structure, and the ensemble- and time-averaged mean square displacements. Some of those properties turn out rather counter-intuitive due to, or irrespective of, possible degeneracy of time-changing mechanisms in the long run. In light of those illustrative comparisons, we examine whether such time-changing operations are worth the additional operational cost, relative to physically relevant characteristics induced, and deduce practical implications and precautions from modeling and inference perspectives, for instance, on the experimental setup involving those anomalous diffusion processes, such as the observation start time and stepsize when measuring mean square displacements, so as to preclude potentially misleading results and paradoxical interpretations.

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.

The influence of nonlinearities on jet noise modeling based on parabolized stability equation

With the aid of a large eddy simulation (LES) model of a turbulent jet, we study the modeling of jet noise based on wavepackets by considering a certain degree of nonlinearity. Linear parabolized stability equations (PSEs) are utilized to solve the spatial evolution of wavepackets with the base flow obtained from the LES. The spectral proper orthogonal decomposition (SPOD) is performed to extract the most energetic coherent structures. Since the mean flow includes partial nonlinearity, improved agreement of hydrodynamic pressure fields between SPOD-filtered results and linear-PSE solutions is obtained in the near field. Deviations only occur when the coherent structures decay. Although linear-PSE solutions represent the near-field hydrodynamics reasonably, the far-field noise propagated from this linear model shows a large deviation from the LES results. Then, a small external harmonic forcing is added to the right-hand side of the PSE to mimic the effects of nonlinearity due to incoherent fluctuations on the late evolution of near-field wavepackets, and an adjoint approach is further utilized to search for optimal forcing distribution. Optimized forcing is mainly located near the critical layer; enhances the energy of wavepackets; and raises the sound radiation efficiency, but to a limited extent. Meanwhile, the coherence-matched PSE wavepacket is proposed to incorporate the coherence decay of wavepackets calculated based on LES. An improved agreement in far-field sound pressure levels for low-frequency components is achieved. In short, these findings all prove the vital role of nonlinearity in jet noise modeling, and the current modeling approaches have made some progress. However, a more physics-based and generalized nonlinear model is still required.

Response Suppression of FGM Plate Using Piezoelectric Layers Under Parametric Uncertainty Conditions with Markovian Jump Approach

It should be noted that in addition to the geometry, constituent material also affects the strength and rigidity of the cylindrical shell, some factors that determine the transient response are its geometry and the constituent material. The capability of piezoelectric materials to adept their properties in reaction to environmental factors including electricity and loading is one of the major reasons for using in this work. Therefore, in this study, the transient response of a symmetric annular sandwich plate incorporating functionally graded core and piezoelectric layers under external harmonic force and electrical voltage is investigated. The properties of the core material vary along its thickness according to a power law model. The displacement field is represented by the third-order shear deformation theory. With the aid of Hamilton’s principle, the structural equations are obtained in terms of displacement components, then solved using the differential quadrature method. In addition, the time response is evaluated with respect to effective parameters including the internal radius, power law index, core thickness, and external voltage. According to the simulation results, the oscillation amplitude decreases as the internal radius of the plate increases over the desired time interval. Also, a higher index parameter is associated with a wider time response range. Moreover, the stability analysis of a piezoelectric system with [Formula: see text] performance is considered based on the theory of Markovian jump systems. To this end, a Markovian jump state-space model of the piezoelectric system obtained using system identification under the effect of external disturbance is presented. The [Formula: see text] stability index is selected based on a candidate Lyapunov function that leads to a set of linear matrix inequalities for each region. The uncontrolled and controlled transient responses of the coupled system under external disturbance are calculated and compared, indicating the satisfactory controller performance in the presence of external disturbance and jump in the sensor and system dynamics.

Long-time evolution of charged grains in plasma under harmonic external force and after being withdrawn

Evolution of the charged grains in a two-dimensional dusty plasma under a spatially harmonic external force, in particular, their long-time behaviors after the force has been withdrawn, is studied by using molecular dynamics simulation. Under an external force and a grain–grain interaction force, initially homogeneously distributed grains can reach a quasi-stationary state in the form of a disk crystal. After the external force is withdrawn, the disk moves initially with its size and shape nearly unchanged until it rapidly stops moving, and eventually the disk grain rotates like a vortex. The time needed to reach the final state increases with the strength of the initial external force increasing.

Wave transmission and vibration response in periodically stiffened plates using a free wave approach.

There are various structures constructed with periodically stiffened thin plates. Vibration prediction of such structures is not easy compared to the structures comprised of uniform plates only due to the mathematical complexity stemmed from the periodic nature. This study provides the analytic method to predict the wave transmission at junctions connecting two semi-infinite periodic structures and the response of a finite periodic structure to an external harmonic point force. The same theoretical framework is employed for dealing with both phenomena. First, free wave solutions are obtained by solving the governing equation for the bending motion of a periodically stiffened, infinite plate using the spatial Fourier Transform and the Floquet's theorem. Then, the free wave solutions are linearly superposed, and the linear coefficients are calculated by applying the appropriate boundary conditions. Numerical simulation is conducted. In dealing with the periodic finite structure, the result is compared with that by the finite element analysis. It is revealed that the periodic nature of the structures affects both the energy transmission and the vibration response of the periodically stiffened plates.

External-field driven motion of charged grains in magnetized plasma

Long-time evolution of charged grains in a two-dimensional magnetized plasma under a spatially harmonic external force is investigated using molecular dynamics simulation. It is found that the grains can converge into a single quasi-steady crystal disc, and their trajectories trace out a complex but regular flowery pattern that gradually becomes simpler in time until a quasi-steady rotating pattern is reached.

Transverse Vibration Energy Harvesting of Double Elastic Steel

This study uses the piezoelectric technology to collect vibration energy from the fixed-fixed nonlinear elastic beams attached with the piezo-patch between the two ends. Both single elastic steel sheet (SESS) and double elastic steel sheet (DESS) systems are investigated and correlated. To simulate the power generation of the vibration energy harvester (VEH) of both the SESS and the DESS in different engineering elements, the simple harmonic external force generated by a shaker at the location of the piezo-patch is used as the source. With this, more vibration converted electric energy is derived from the transverse deformation and flapping from the DESS than the SESS beam. The equation of a nonlinear Euler–Bernoulli beam is coupled with the electric energy equation of the piezo-patch to simulate the SESS VEH system. The flapping force from the DESS VEH system can be considered the concentrated external load applied on the SESS beam model. The method of multiple scales (MOMS) is employed to analyze this nonlinear problem. The fixed points plots and the numerical results confirm this theory presented for the two beam systems, which can be used for evaluating similar engineering systems. Experiments are also performed in this study. The Taguchi method is used to analyze the optimum locations of the shaker and piezo-patch, as well as the confidence level of the factors. The method of nonlinear analysis presented in this study demonstrates its accuracy compared with the linear case. The transverse DESS VEH model proposed is proved to be feasible and more effective than the SESS system.

Исследование установившихся периодических движений модулей ползающего робота при наличии внешних возмущений

The development and design of a multi-module crawling robot requires an elaboration of each aspect of its motion in the implementation of various tasks, including holding the position of such a robot under a driving force. This problem has been solved for a three-module crawling robot, one of the end supports of which was fixed on the surface, and the other was affected by the harmonic external force changing according to the meander law. For this purpose, a module control system based on the CTC controller has been proposed. Numerical simulation has been used to study the quantitative and qualitative characteristics of robot module vibrations depending on the ratio of the amplitudes of the vertical and horizontal components of the force and the phase mismatch between them. It is found that when the external force vertical and horizontal amplitudes are equal, the module angles fluctuate relative to the set values, and when there is a certain ratio between the amplitudes, a static error appears which increases as the specified ratio moves away from 1.

Emergence and synchronization of a reversible core in a system of forced adaptively coupled Kuramoto oscillators.

We report on the phenomenon of the emergence of mixed dynamics in a system of two adaptively coupled phase oscillators under the action of a harmonic external force. We show that in the case of mixed dynamics, oscillations in forward and reverse time become similar, especially at some specific frequencies of the external force. We demonstrate that the mixed dynamics prevents forced synchronization of a chaotic attractor. We also show that if an external force is applied to a reversible core formed in an autonomous case, the fractal dimension of the reversible core decreases. In addition, with increasing amplitude of the external force, the average distance between the chaotic attractor and the chaotic repeller on the global Poincaré secant decreases almost to zero. Therefore, at the maximum intersection, we see a trajectory belonging approximately to a reversible core in the numerical simulation.

Impedance based service life assessment of corroded structures with Cross correlation analysis

Electro mechanical impedance technique has garnered major significance in structural health monitoring field by integrating with information and sensing technology. Technical advancements in automated smart devices improved the diagnostic evaluation of this monitoring method by incorporating piezoelectric materials. The optimistic coupling property of these materials enhanced the life span of structural members by detecting damages at incipient stage. The intervention of piezoelectric transducers has created potential applications for impedance approach in assessing the structural performance with high feasibility. The present work aims at estimating the service life of simply supported smart beam using piezoelectric sensors in Abaqus Simulia software. Corrosion is deployed at a uniform rate in reinforced concrete beam with impressed current flow and the loss in diameter of rebar is calculated using Faraday’s electrochemical equation. With frequency domain based impedance technique numerical analysis is conducted on the modeled beams by exciting the sensor with an external harmonic force and the output electric responses of different piezoelectric locations are captured. The steady state dynamic responses for different cases are compared and quantified with a statistical damage algorithm in MATLAB. The resulted cross correlation indices are analyzed to prevent further deterioration of the corroded beams by setting a limit value. The proposed simulation technology is capable of estimating the residual life span of damage structures with cross correlation analysis.

РЕАКТАНСЫ И САССЕПТАНСЫ МЕХАНИЧЕСКИХ СИСТЕМ

The classical solution to the problems associated with calculating the velocities and reactions of elements of complex mechanical systems under harmonic force consists in the compilation and integration of systems of differential equations and is rather cumbersome and time-consuming. In most cases, a steady state is of major interest. The purpose of this study is to develop essentially compact methods for calculating systems under steady-state conditions. The problem is solved by the methods which are typically used to calculate electrical circuits. Representation of harmonic quantities as rotating vectors in a complex plane and the operations with their complex amplitudes can greatly facilitate the calculation of arbitrarily complex mechanical systems under harmonic effects in the steady state. In the proposed method, a key role is played by mechanical reactance, resistance, and impedance for the parallel connection of consumers of mechanical power, as well as susceptance, conductance, and admittance for the serial one. At force resonance, the total reactance of the mechanical system is zero. This means that the system does not exhibit reactive resistance to the external harmonic force. At velocity resonance, the total susceptibility of the mechanical system is zero. This means that the system has infinitely high resistance to the external harmonic force. As a result, the stock of the source of harmonic force is stationary, although the inert body and the elastic element oscillate.

RELAXATION OF EDGE DISLOCATIONS IN METALS UNDER THE ACTION OF EXTERNAL STRESSES

The study of the relaxation of edge dislocations (ED) in metals under the action of alternating external stresses of various natures is of great practical interest for estimating of workability of structural elements and functional materials of nuclear power and thermonuclear fusion power plants. In the model of the inhomogeneous dissipative sine – Gordon equation for different values of the amplitude and frequency of the external harmonic force, as well as for different values of dislocation friction, the nonlinear dynamics of ED with fixed ends is studied numerically. The formation of breather-type solitons on the ED is shown, the maximum amplitude of which decreases with increasing dislocation friction. The resonance dependence of the kinetic energy of ED oscillations on the frequency of the external field is described. It has a maximum value for the resonant frequency and decreases with increasing numerically found resonance frequencies.

Reaction-Diffusion Equation with Stationary Wave Perturbation in Weakly Ionized Plasmas

The influence of external harmonic forcing in nonlinear chaotic systems is a subject of active investigation, chiefly in low-dimensional dynamical systems, but fewer results are available for high-dimensional systems like plasmas. In this paper, we consider a theoretical model for a weakly ionized plasma, in which the following effects are taken into account: (i) ambipolar diffusion; (ii) the inflow of plasma particles through ionization processes; (iii) the outflow of plasma particles due to recombination. The spatiotemporal patterns of the resulting nonlinear system, as revealed by numerical integration of the reaction-diffusion partial differential equations, can be partially or totally suppressed through the action of acoustic waves forming stationary patterns in the plasma container. These suppression effects are quantitatively investigated by a number of numerical diagnostics like the average Lyapunov exponents and spatial correlation function. Suppression of spatiotemporal chaos can be achieved through adequate choices of amplitude and frequency of the applied waves.

Nonlinear Resonant Responses, Mode Interactions, and Multitime Periodic and Chaotic Oscillations of a Cantilevered Pipe Conveying Pulsating Fluid under External Harmonic Force

The nonlinear resonant responses, mode interactions, and multitime periodic and chaotic oscillations of the cantilevered pipe conveying pulsating fluid are studied under the harmonic external force in this research. According to the nonlinear dynamic model of the cantilevered beam derived using Hamilton’s principle under the uniformly distributed external harmonic excitation, we combine Galerkin technique and the method of multiple scales together to obtain the average equation of the cantilevered pipe conveying pulsating fluid under 1 : 3 internal resonance and principal parametric resonance. Based on the average equation in the polar form, several amplitude-frequency response curves are obtained corresponding to the certain parameters. It is found that there exist the hardening-spring type behaviors and jumping phenomena in the cantilevered pipe conveying pulsating fluid. The nonlinear oscillations of the cantilevered pipe conveying pulsating fluid can be excited more easily with the increase of the flow velocity, external excitation, and coupling degree of two order modes. Numerical simulations are performed to study the chaos of the cantilevered pipe conveying pulsating fluid with the external harmonic excitation. The simulation results exhibit the existence of the period, multiperiod, and chaotic responses with the variations of the fluid velocity or excitation. It is found that, in the cantilevered pipe conveying pulsating fluid, there are the multitime nonlinear vibrations around the left-mode and the right-mode positions, respectively. We also observe that there exist alternately the periodic and chaotic vibrations of the cantilevered pipe conveying pulsating fluid in the certain range.

RETRACTED ARTICLE: Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM

In this research, a mathematical derivation is made to develop a nonlinear dynamic model for the nonlinear frequency and chaotic responses of the multi-scale hybrid nano-composite reinforced disk in the thermal environment and subject to a harmonic external load. Using Hamilton’s principle and the von Karman nonlinear theory, the nonlinear governing equation is derived. For developing an accurate solution approach, generalized differential quadrature method (GDQM) and perturbation approach (PA) are finally employed. Various geometrically parameters are taken into account to investigate the chaotic motion of the viscoelastic disk subject to harmonic excitation. The fundamental and golden results of this paper could be that in the lower value of the external harmonic force, different FG patterns do not have any effects on the motion response of the structure. But, for the higher value of external harmonic force and all FG patterns, the chaos motion could be seen and for the FG-X pattern, the chaosity is more significant than other patterns of the FG. As a practical designing tip, it is recommended to choose plates with lower thickness relative to the outer radius to achieve better vibration performance.

Forced Oscillations of the Discrete Membrane Under Conditions of “Sonic Vacuum”

Abstract This study presents a new analytical model for nonlinear dynamics of a discrete rectangular membrane that is subjected to external harmonic force. It has recently been shown that the corresponding autonomous system admits a series of nonlinear normal modes. In this paper, we describe stationary and non-stationary dynamics on a single mode manifold. We suggest a simple formula for the amplitude-frequency response in both conservative and non-conservative cases and present an analytical expression (in parametric space) for thresholds for all possible bifurcations. Theoretical results obtained through asymptotic approach are confirmed by the experimental data. Experiments on the shaking table show that amplitude-frequency response to external force in a real system matches our theory. Substantial hysteresis is observed in the regimes with increasing and decreasing frequency of external force. The obtained results may be used in designing nonlinear energy sinks.

Stochastic multiresonance in oscillators induced by bounded noise

Stochastic resonance (SR) in a harmonic oscillator with parametric bounded noise and external periodic force is investigated. By applying the property of bounded noise and using Cameron–Martin formula, infinite chains of linear differential equations are obtained for the spectral amplification, mean-square displacement, and variance. Multiple stochastic resonances are observed for these characteristics. Relationships with resonant tongues of the Mathieu equation are established. It is shown that similar effects can be observed in the case of harmonic noise external force. The case of two coupled oscillators is also studied. Analysis is based on a natural truncation of the infinite moment chains and using of the numerical matrix computations.

New Indices for Evaluating Vibration Characteristics of Flexible-Joint Robots

Structural vibration is a significant consideration for robotic applications such as machining where the robot is subject to large dynamic loading. Aiming at providing an efficient means to evaluating the vibration characteristics of industrial robots for these applications, this work proposes two new indices to quantify the elastic displacement of the tool mounted on the robot caused by the vibrations induced by external process loading for flexible-joint robots. For this purpose, a structural dynamic model is first developed to derive the frequency responses of the tool displacement. Then, the displacement-force and displacement-torque frequency response ratios are defined, which represent the mapping from the amplitudes of an external harmonic force and torque to the amplitude of tool displacement respectively. The upper bounds of the two ratios are used as evaluation indices for the vibration characteristics of the robot, which represent the worst situation of the tool displacement due to harmonic excitation with amplitude of unit force and unit torque respectively. With these indices, an efficient method is provided to predict whether the tool misalignment caused by periodic loading is acceptable for process quality requirement. Numerical simulation demonstrates the effectiveness of the proposed method for a robotic riveting system being developed for aerospace assembly.