Amplitude Of Harmonic Excitation Research Articles

Nonlinear vibration and acoustic radiation of an internally resonant buckled beam

Although the nonlinear vibration of buckled beam has been thoroughly studied, its acoustic radiation does not get much attention. This paper deals with the nonlinear vibro-acoustic problem of an internally resonant buckled beam immersed in an infinite fluid. A finite element method based on an arbitrary Lagrangian-Eulerian framework is adopted to analyze the nonlinear interaction between the large-deformed buckled beam and the finite-amplitude acoustic waves in the fluid. The effects of acoustic pressure excitation, external harmonic excitation, and internal resonance on the vibro-acoustic responses of the buckled beam are discussed in the first and second primary resonances. It is found that the amplitude of the acoustic pressure can be comparable to the amplitude of the external harmonic force. This leads to an increase in the critical harmonic excitation amplitude for the occurrence of quasi-periodic responses and an increase in the second mode frequency component of quasi-periodic responses compared to the case neglecting the acoustic pressure excitation on the beam. Saturation phenomena are discovered in the second mode frequency components of beam displacement and acoustic pressure. Due to the 2:1 internal resonance, different time-frequency characteristics, quadrupole acoustic directivity patterns with low radiation efficiency, and dipole acoustic directivity patterns with high radiation efficiency can be observed at different harmonic excitation amplitudes and frequencies. Understanding these phenomena will help design loudspeakers and recognize sound sources.

Nonlinear Vibration of Truncated Open Conical Nanoshells Under Harmonic Excitation

The nonlinear forced vibration behavior of truncated open conical nanoshells is investigated in this paper. Nonlocal elasticity theory has been employed to modify the size-dependent effect of micro/nanostructure. Von Karman nonlinear strains are used to consider the geometric nonlinearity of the nanoshells. Using Hamilton’s principle, the equations of motion and boundary conditions of truncated open conical nanoshells are derived. Galerkin method is used to solve the governing equations. Then, to determine the nonlinear forced vibration behavior of truncated open conical nanoshells, complex averaging and arc-length continuation methods are employed. Finally, the effect of nonlocal parameters, cone apex angle, the amplitude of harmonic excitation, structural damping coefficient, and geometrical parameters on the nonlinear frequency response of the truncated open conical nanoshells are discussed.

Research on the optimal excitation threshold interval of targeted energy transfer for piezoelectric nonlinear energy sink

In this paper, in order to design a piezoelectric nonlinear energy sink system (NES), a coupling system of elastic plate and piezoelectric nonlinear energy sink is established. The nonlinear stiffness and damping are realized by the shunt circuit of piezoelectric materials. By using the harmonic balance method to analyze the forced vibration characteristics of the coupling system, it is found that within a certain range of harmonic excitation amplitude, the targeted energy transfer (TET) phenomenon occurs in the system. There is a threshold interval for the TET phenomenon of the system, where the TET does not occur when the excitation is less than the lower limit of the threshold. When the excitation exceeds the upper limit of the threshold, the system response appears a new high amplitude branch. The definite solution to the lower limit of the excitation threshold is determined, but when the system parameters are different, the high amplitude branch of the system response will appear above or below the amplitude of the platform. The results of this paper give the solution of the optimal TET excitation threshold interval in these two different cases and are helpful to design a NES for different applications.

Time-varying dynamic analysis of a plate entering the finite subsonic airflow field

Devices entering rigid wall constrained finite airflow field are commonly seen in engineering practice. The devices gradually move from the infinite airflow field to the finite airflow field, which makes the device and the outside flow field a complicated time-varying fluid–structure interaction system. During the entry process, the stiffness, damping and mass of the devices change constantly over time, which makes the dynamic behaviors of the system show complex time-varying properties. Hence, it is needed to study the stability and vibration characteristics of the devices entering the finite airflow field. In this study, based on the von Karman nonlinear plate theory and the linear potential flow theory, the governing equation of the plate entering the finite airflow field is established by employing the Hamilton’s principle. By using the generalized eigenvalue method to linear system, the variations of the natural frequencies of the plate in the process of entering the finite airflow field are investigated, from which the stability change rules of the system are obtained. The vibration properties of the system are studied by analyzing the displacement time response of the plate in the process of entering the finite flow field. From the results, the different system parameters such as entry position, entry velocity, aspect ratio, external harmonic excitation amplitude and external harmonic excitation frequency have significant impacts on the stability and vibration properties of the plate entering the finite airflow field.

Stochastic dynamic analysis of rolling ship in random wave condition by using finite element method

Nonlinear stochastic rolling is a primary contributor to ship instability. Random wave excitation is described in this study as a combination of harmonic excitation and Gaussian white noise excitation, and the nonlinear rolling of a ship subjected to this composited excitation is investigated using stochastic vibration methodology and a numerical analysis approach. The Fokker-Planck (FP) equation for nonlinear stochastic ship rolling is derived, and the associated transient probability density function (PDF) is numerically solved by applying the finite element method (FEM) and Crank-Nicolson method, and the findings are manifested to be consistent with the Monte Carlo simulation (MCS). This proves the applicability and effectiveness of the FEM in the numerical study of the nonlinear stochastic ship rolling. Then the effects of harmonic excitation amplitude and stochastic excitation intensity in stable and unstable regions on nonlinear ship rolling are investigated, which provide essential references for ship stability and capsizing research.

On the Effect of the Electrical Load on Vibration Energy Harvesting Under Stochastic Resonance

Abstract Vibration energy harvesting (VEH) is a promising alternative for powering wireless electronics in many practical applications. Ambient vibration energy in the surrounding space of a target application often involves an inescapable randomness in the exciting vibrations, which may lead to deterioration of the expected power gains due to insufficient tuning and limited optimal designs. Stochastic resonance (SR) is a concept that has recently been considered for exploiting this randomness toward improving power generation from vibrating systems, based on the coexistence of near-harmonic vibrations with broadband noise excitations in a variety of practical mechanical systems. This paper is concerned with the optimal conditions for SR in vibration energy harvesters, exploring the frequently neglected effect of realistic architectures of the electrical circuit on the system dynamics and the achievable power output. A parametric study is conducted using a numerical path integration (PI) method to compute the response probability density functions (PDFs) of vibration energy harvesters, focusing on the effect of standard electrical components; namely, a load resistor, a rectifier, and a capacitor. It is found that the conditions for SR exhibit a nonlinear dependence on the weak harmonic excitation amplitude. Moreover, the modified nonlinear dissipation properties introduced by the rectifier and the capacitor lead to a tradeoff between the power output and the nonconducting dynamics that is essential in order to determine optimal harvesting designs.

Global analysis of crises in shape memory thin plate system

The unique global properties of shape memory alloy are mainly derived from the martensite phase transition and its inverse, which result from the change of temperature and external load. In this paper, the global characteristics of shape memory alloy thin plate system are analyzed with the temperature and harmonic excitation amplitude as control parameters. Based on the method of Poincare map, the complex crisis phenomenon of the system including the sudden change in number, size and type of attractors can be observed through the global multivalued bifurcation diagram. However, the specific crisis type is not clear, it is necessary to be analyzed from the global viewpoint. By computing the global diagram with the composite cell coordinate system method which constructs a composite cell state space by multistage division of the continuous phase space, the attractors, saddles and basins of attraction of the system can be obtained more accurately. The vivid evolutionary processes of the crisis phenomena of the system are illustrated, and it can be found that the system presents a complex global structure with amplitude and temperature changing. There exist two kinds of crises: one is the boundary crisis resulting from the collision between a chaotic/periodic attractor and a chaotic saddle within the basin boundary, which causes the attractor to vanish, and the other is the merging crisis caused by the collision of two or more attractors with the chaotic saddle within the basin boundary where a new chaotic attractor appears. When multiple attractors coexist in the system, the basin boundary may be smooth or fractal, and for any point at boundary, its small open neighborhood always has a nonempty intersection with three or more basins, which is known as Wada basin boundary. It is difficult to predict the dynamic behavior of the system accurately due to the fractal, the Wada-Wada, Wada-fractal and fractal-Wada basin boundary metamorphoses which can be observed along with the variation of temperature and amplitude through the composite cell coordinate system method, which owns a unique advantage in depicting basin boundary. Furthermore, the Wada property is displayed more clearly by refining specified region. The results of this paper provide a theoretical analysis tool for adjusting the dynamic response of shape memory alloy thin plate system and optimizing the deformation and vibration control of mechanical equipment through controlling temperature and excitation intensity.

Primary and Secondary Resonance of Micro-resonators Based on Couple Stress Theory

In this work, the influence of size effect on the primary and the secondary resonance of micro-resonators is investigated. The micro-resonator, which is considered in this study, is a clamped–clamped beam excited by AC and DC electrostatic voltage. The DC voltage is applied to prevent doubling of excitation frequency and tune the range of system’s working frequency. We computed the beam’s motion equation using Euler–Bernoulli theory for wide beams considering couple stress theory and geometrical nonlinearity. For micro-resonators with high quality factor and small amplitude of harmonic excitation, we captured frequency response curves using shooting method, then we used multiple time scale method for its validation. Due to the limitation of the multiple time scale perturbation method for the computation of frequency responses in large amplitude harmonic excitation, the shooting method has been used, regardless of how big the harmonic excitation amplitude or how small the micro-resonator’s quality factor is. Another contribution of this paper is the analysis of the influence of size effect on the system’s hardening and softening as well as its effect on reduction of pull-in band. This work presents that considering the size effect will result in reduction or elimination of pull-in band and also the nonlinear behavior of the system will be inclined to become linear in primary resonance.

Characterizing the robustness and susceptibility of steady-state dynamics in post-buckled structures to stochastic perturbations

Recent reports exemplify both the potential and the concerns encountered in implementing post-buckled structures in diverse applications. By numerical and experimental methods, characterizations have provided useful insights on the dynamic sensitivities of such bi- or multistable structures subjected to either harmonic or stochastic excitations. Yet, the more realistic scenario of combined harmonic and stochastic loading has been not been closely examined so that such sensitivities might be fully illuminated. To provide a more complete understanding on the robustness and susceptibility of multistable structures to dynamic state transitions, this research establishes new analytical and experimental methods to quantify the likelihood of triggering transitions among dynamic regimes of an archetypal post-buckled structure as a result of combined harmonic and stochastic loading. It is discovered that persistent periodic snap-through dynamics are rapidly disabled by additional noise excitation when the harmonic excitation contribution occurs at frequencies close to the linearized resonance. The extra noise may also drastically compromise the integrity of small-amplitude periodic responses that occur at frequencies around one-half of the linearized resonance. Particular relative proportions of the noise standard deviation and harmonic excitation amplitude are uncovered that most readily compromise the robustness of a given steady-state dynamic regime. The analytical method also complements prior developments focused on stochastic resonance by uncovering the broader perspective of dynamic sensitivities of post-buckled structures under arbitrary combinations of harmonic and random driving loads.

Stochastic resonance in a genetic toggle model with harmonic excitation and Lévy noise

Stochastic resonance is investigated to explain the beneficial effect of Lévy noise on gene expression of genetic toggle model with harmonic excitation. The dynamic change of protein concentration of genetic toggle model under combined drives of harmonic excitation and Lévy noise is obtained numerically. Stochastic resonance is presented through the classical measure of signal-to-noise-ratio. Then from two aspects of combined drives on the protein at high or low concentration, the changes of protein concentration and signal-to-noise-ratio are discussed, respectively. When combined drives are within the protein at high concentration, the increasing Lévy noise intensity can promote the transition between the high and low concentrations, and the low protein concentration hardly fluctuates under the small noise intensity. It is also shown that the increase of stability index, skewness parameter of Lévy noise and amplitude of harmonic excitation can suppress the optimum collaboration of stochastic resonance. On the other hand, when combined drives are within the protein at lowconcentration, the increasing noise intensity can enhance the transition between the high and low concentrations, and the increase of stability index, skewness parameter and amplitude can strengthen the optimum collaboration of stochastic resonance. By the synergic actions of stochastic resonance, it is demonstrated that combined effect of harmonic excitation and Lévy stimuli can be utilized to promote the gene expression of proteins in genetic toggle model.

Hysteresis-induced bifurcation and chaos in a magneto-rheological suspension system under external excitation**Projects supported by the National Natural Science Foundation of China (Grant Nos. 51475246, 51277098, and 51075215), the Research Innovation Program for College Graduates of Jiangsu Province China (Grant No. KYLX15 0725), and the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20131402).

The magneto-rheological damper (MRD) is a promising device used in vehicle semi-active suspension systems, for its continuous adjustable damping output. However, the innate nonlinear hysteresis characteristic of MRD may cause the nonlinear behaviors. In this work, a two-degree-of-freedom (2-DOF) MR suspension system was established first, by employing the modified Bouc–Wen force–velocity (F–v) hysteretic model. The nonlinear dynamic response of the system was investigated under the external excitation of single-frequency harmonic and bandwidth-limited stochastic road surface. The largest Lyapunov exponent (LLE) was used to detect the chaotic area of the frequency and amplitude of harmonic excitation, and the bifurcation diagrams, time histories, phase portraits, and power spectrum density (PSD) diagrams were used to reveal the dynamic evolution process in detail. Moreover, the LLE and Kolmogorov entropy (K entropy) were used to identify whether the system response was random or chaotic under stochastic road surface. The results demonstrated that the complex dynamical behaviors occur under different external excitation conditions. The oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations, and chaotic oscillations was observed in detail. The chaotic regions revealed that chaotic motions may appear in conditions of mid-low frequency and large amplitude, as well as small amplitude and all frequency. The obtained parameter regions where the chaotic motions may appear are useful for design of structural parameters of the vibration isolation, and the optimization of control strategy for MR suspension system.

Nonlinear analysis of energy harvesting systems with fractional order physical properties

An electromechanical energy harvesting system with a fractional order current–voltage relationship for the electrical circuit and fractional power law in the restoring force of its mechanical part is considered to act as an energy harvester. Our results show that, under a single-well potential configuration, for a small amplitude of the perturbation, as the order of derivative increases, the resonant amplitude of mechanical vibration decreases while the bending degree (hardening case) remains fairly constant. For a large amplitude of the perturbation, the output power is increased due to the hardening effects. Under a double-well configuration, the fractional power stiffness \(\alpha \) strongly affects the crossing well dynamics (large amplitude motion) and consequently the output electrical power. The harvested electric power appears to be maximal for deterministic and random excitation for small \(\alpha \). High-level noise intensity is found to reduce the output power in the region of resonance and surprisingly increases the output in other regions of \(\alpha \). For sufficiently large amplitude of harmonic excitation, this effect is realized in a stochastic resonance.

Mechanism for the coexistence phenomenon of random phase suppressing chaos and stochastic resonance in Duffing system

Noise, which is ubiquitous in real systems, has been the subject of various and extensive studies in nonlinear dynamical systems. In general, noise is regarded as an obstacle. However, counterintuitive effects of noise on nonlinear systems have recently been recognized, such as noise suppressing chaos and stochastic resonance. Although the noise suppressing chaos and stochastic resonance have been studied extensively, little is reported about their relation under coexistent condition. In this paper by using Lyapunov exponent, Poincaré section, time history and power spectrum, the effect of random phase on chaotic Duffing system is investigated. It is found that as the intensity of random phase increases the chaotic behavior is suppressed and the power response amplitude passes through a maximum at an optimal noise intensity, which implies that the coexistence phenomenon of noise suppressing chaos and stochastic resonance occurs. Furthermore, an interesting phenomenon is that the optimal noise intensity at the SR curve is just the critical point from chaos to non-chaos. The average effect analysis of harmonic excitation with random phase and the system’s bifurcation diagram shows that the increasing of random phase intensity is in general equivalent to the decreasing of harmonic excitation amplitude of the original deterministic system. So there exists the critical noise intensity where the chaotic motion of large range disintegrates and non-chaotic motion of small scope appears, which implies the enhancing of the regularity of system motion and the increasing of the response amplitude at the input signal frequency. After that, the excess noise will not change the stability of the system any more, but will increase the degree of random fluctuation near the stable motion, resulting in the decreasing of the response amplitude. Therefore, the formation of stochastic resonance is due to the dynamical mechanism of random phase suppressing chaos.

Tuning the primary resonances of a micro resonator, using piezoelectric actuation

Microbeam dynamics is important in MEMS filters and resonators. In this research, the effect of piezoelectric actuation on the resonance frequencies of a piezoelectrically actuated capacitive clamped-clamped microbeam is studied. The microbeam is sandwiched with piezoelectric layers throughout its entire length. The lower piezoelectric layer is exposed to a combination of a DC and a harmonic excitation voltage. The DC electrostatic voltage is applied to prevent the doubling of the excitation frequency. The traditional resonators are tuned using DC electrostatic actuation, which tunes the resonance frequency only in backward direction on the frequency domain. The proposed model enables tuning the resonance frequencies in both forward and backward directions. For small amplitudes of harmonic excitation and high enough quality factor, the frequency response curves obtained by the shooting method are validated with those of the multiple time scales technique. Unlike the perturbation technique, which imposes limitation on both the amplitude of the harmonic excitation and the quality factor to be applicable, the shooting method can be applied to capture the periodic attractors regardless of how big the amplitude of harmonic excitation and the quality factor are.

Annihilation of high-amplitude periodic responses of a forced two degrees-of-freedom oscillatory system using nonlinear energy sink

This paper is devoted to the investigation of the role of a single degree-of-freedom (d.o.f) nonlinear energy sink (NES) with nonlinear damping characteristics in annihilating undesired periodic response regimes and simultaneously preserving strongly modulated responses (SMR) in a harmonically excited three d.o.f system consisting of two linear coupled oscillators and an NES attached to it. The NES parameters are tuned such that they are able to eliminate all undesired periodic branches and to have the SMR region in the vicinity of both modes of the system concurrently. The tuning procedure leads to a maximum SMR interval which is achievable using this type of the NES in the vicinity of each excited mode. According to the results, design requirements are achieved using this type of the NES for a two d.o.f primary system for the first time. But, further increase in the d.o.f of the system will lead to a restriction of the range of harmonic excitation amplitude of the first oscillator for which the design requirements would be satisfied.

Nonlinear vibrations and chaos in electrostatic torsional actuators

Electrostatic torsional micro-mirrors have wide spread use in different industries for diverse purposes. This paper investigates the development of superharmonics and chaotic responses in electrostatic torsional micro-mirrors near the pull-in condition. Appearance of nonlinear phenomena is investigated in models accounting for and disregarding the coupling of torsional and flexural deflections. Analysis of the system response to step and harmonic excitation reveals the appearance of DC and AC symmetry breaking. Increasing the amplitude of harmonic excitation, the response in the form of distinct superharmonics changes to a broad band response, where there is loss of periodicity and the response becomes chaotic. Accounting for flexural deflections in coupled model reduces the voltage thresholds corresponding to symmetry breaking and chaotic responses. It is also shown that damping has a regularizing effect and introduction of damping changes the chaotic undamped response into quasi-periodic one.

Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory

This paper deals with problems for which it is necessary to represent the oscillations of the electromechanical seismograph about their position of equilibrium as regards the synchronous condition. The loss of stability of the system occurs through a saddle-node bifurcation, where there is a collision of the stable orbit with an unstable one. Then, global bifurcations and chaotic dynamics of an electromechanical seismograph are the aims of this study. The electrical part of the model is described by an extended force Rayleigh oscillator with Φ6-potential, while the mechanical part is described by a damped and driven linear oscillator. By using the direct perturbation technique, we analytically obtain the general solution of the first-order equation. Through the boundedness condition of the general solution we get the famous Melnikov function predicting the onset of chaos in the case where the Φ6-potential is three wells, which are complemented by numerical simulations by which we illustrate the bifurcation curves and the fractality of the basins of attraction. The results show that the threshold amplitude of harmonic excitation for the onset of instability will move upwards as the amplitude intensity of the ground motion increases. These results suggest that much attention should be paid to controlling the increase of the amplitude of the ground motion, especially when the harmonic excited electromechanical seismograph system as a main device is applied to some practical systems.

EFFECT OF GAUSSIAN WHITE NOISE ON THE DYNAMICAL BEHAVIORS OF AN EXTENDED DUFFING–VAN DER POL OSCILLATOR

The influence induced by random noise on dynamical behaviors is a classical yet challenging subject. This paper discusses the influence of Gaussian white noise on the dynamics of a self-excited triple well extended Duffing–Van der Pol oscillator already subjected to harmonic excitation. Firstly, the condition for the rise of hom/heteroclinic chaos is derived by random Melnikov's technique under its corresponding mean-square criterion and the result indicates that the threshold amplitude of harmonic excitation is lowered by the appearance of Gaussian white noise. Moreover, the threshold is decreased as the noise intensity increases. Since the Melnikov's criterion is only a necessary condition for the occurrence of chaotic motion, this prediction is tested against numerical simulations of the basins of attraction and the Lyapunov exponents. By vanishing the largest Lyapunov exponents, another criterion for the onset of chaos is obtained which is accorded with the theoretical one. Finally, how the noise effects the structure of periodic or chaotic attractor is investigated by simulating Poincare maps of the original system and rich transition states displayed by the considered extended Duffing–Van der Pol oscillator are observed.

Effect of random noise on chaotic motion of a particle in a ϕ6 potential

Abstract The chaotic behaviors of a particle in a triple well ϕ6 potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated.

Hilbert spectrum and intrinsic oscillation mode of dynamic response of a bilinear SDOF system: influence of harmonic excitation amplitude

Under harmonic wave excitation, the dynamic response of a bilinear SDOF system can be expressed by the Hilbert spectrum. The Hilbert spectrum can be formulated by (1) the inter-wave combination mechanism between the steady response and the transient response when the system behaves linearly, or (2) the intra-wave modulation mechanism embedded in one intrinsic mode function (IMF) component when the system behaves nonlinearly. The temporal variation of the instantaneous frequency of the IMF component is consistent with the system nonlinear behavior of yielding and unloading. As a thorough study of this fundamental structural dynamics problem, this article investigates the influence of the amplitude of the harmonic wave excitation on the Hilbert spectrum and the intrinsic oscillatory mode of the dynamic response of a bilinear SDOF system.