Large-amplitude Response Research Articles

Flexural–torsional modal interaction in MEMS actuators initiated by minuscule asymmetry

AbstractAn efficient actuation technique for electrostatic MEMS actuators exploiting electro-mechanical-mechanical modal interactions is proposed. The flexural–torsional equations of motion are established, and we manifest that the initiation of a 2:1 autoparametric modal interaction between in-plane bending and torsional modes of the actuator that is supposed to be symmetrical with respect to its axis of rotation is contingent upon the presence of a quadratic stiffness term, which arises from the existence of non-zero first moments of area of the actual cross-section in prismatic microbeams. In order to efficiently reduce the AC voltage value required to reach the activation of the 2:1 mechanical modal interaction, the electrical resonant frequency is syntonized to half of the natural frequency of the in-plane bending mode. The results indicate that the amplitude of the in-plane motion saturates upon the initiation of an energy exchange between the bending and torsional motions. Through suitable tuning of the AC frequency, the amplitude of the in-plane motion is minimized, while the amplitude of the torsional motion, the indirectly excited mode, is maximized. Our results demonstrate that the actuator's torsional motion, when subjected to a 1:2:1 electro-flexural–torsional modal interactions, is triggered by applying a maximum voltage of 10 V, resulting in about 20 degrees rotational angle. Furthermore, prolific frequency combs are generated as a result of secondary Hopf bifurcations along the large-amplitude response branches, inducing quasi-periodicity in the MEMS dynamics.

Functional Dynamics and Selectivity of Two Parallel Corticocortical Pathways from Motor Cortex to Layer 5 Circuits in Somatosensory Cortex.

In the rodent whisker system, active sensing and sensorimotor integration are mediated in part by the dynamic interactions between the motor cortex (M1) and somatosensory cortex (S1). However, understanding these dynamic interactions requires knowledge about the synapses and how specific neurons respond to their input. Here, we combined optogenetics, retrograde labeling, and electrophysiology to characterize the synaptic connections between M1 and layer 5 (L5) intratelencephalic (IT) and pyramidal tract (PT) neurons in S1 of mice (both sexes). We found that M1 synapses onto IT cells displayed modest short-term depression, whereas synapses onto PT neurons showed robust short-term facilitation. Despite M1 inputs to IT cells depressing, their slower kinetics resulted in summation and a response that increased during short trains. In contrast, summation was minimal in PT neurons due to the fast time course of their M1 responses. The functional consequences of this reduced summation, however, were outweighed by the strong facilitation at these M1 synapses, resulting in larger response amplitudes in PT neurons than IT cells during repetitive stimulation. To understand the impact of facilitating M1 inputs on PT output, we paired trains of inputs with single backpropagating action potentials, finding that repetitive M1 activation increased the probability of bursts in PT cells without impacting the time dependence of this coupling. Thus, there are two parallel but dynamically distinct systems of M1 synaptic excitation in L5 of S1, each defined by the short-term dynamics of its synapses, the class of postsynaptic neurons, and how the neurons respond to those inputs.

Anatomy of the olfactory system and potential role for chemical communication in the sound-producing Lusitanian toadfish, Halobatrachus didactylus.

The current study investigated the structure and function of the olfactory system of the Lusitanian toadfish, Halobatrachus didactylus, using histology and electrophysiology (electro-olfactogram [EOG]), respectively. The olfactory system consists of a digitated anterior peduncle, of unknown function, containing the inhalant nostril. This then leads to a U-shaped olfactory chamber with the olfactory epithelium-identified by Gαolf-immunoreactivity-on the ventral surface. A large lacrimal sac is connected to this tube and is likely involved in generating water movement through the olfactory chamber (this species is largely sedentary). The exhalent nostril lies by the eye and is preceded by a bicuspid valve to ensure one-way flow of water. As do other teleosts, H. didactylus had olfactory sensitivity to amino acids and bile acids. Large-amplitude EOG responses were evoked by fluid from the anterior and posterior testicular accessory glands, and bile and intestinal fluids. Anterior gland and intestinal fluids from reproductive males were significantly more potent than those from non-reproductive males. Male urine and skin mucus proved to be the least potent body fluids tested. These results suggest that chemical communication-as well as acoustic communication-may be important in the reproduction of this species and that this may be mediated by the accessory glands and intestinal fluid.

Vector form intrinsic finite element analysis for nonlinear parametric resonances of planar beam structures

The parametric resonance of a beam structure may lead to a disastrous consequence. To assess the safety of a structure, structural engineers need a straightforward and practical method to calculate the large-amplitude nonlinear parametric resonance responses of the beam structure. This paper proposes a newly developed vector form intrinsic finite element (VFIFE) calculation scheme to analyse the nonlinear parametric resonance of planar beam structures. By taking into account the geometric stiffness effect of the internal axial force for the frame element, a new relationship between internal nodal forces and deformation components is first established based on the VFIFE principle and the parametric resonance mechanism of frame structures. A numerical example of a portal frame is presented to demonstrate the efficacy of this new calculation scheme. A parametric resonance test of the cantilever beam is conducted to verify the applicability of the VFIFE method. The numerical results show that the proposed scheme can well simulate the nonlinear phenomena of parametric resonances. The numerical predictions are consistent with the experimental observations. A filtering-based finite-time Lyapunov exponent approach is presented to determine the stability boundary of the structure. The numerical stability boundary of parametric resonance agrees well with the experimental boundary. Possible causes of deviations between the numerical and experimental results are discussed. The proposed VFIFE scheme is a simple and effective means of analysing the stability and nonlinear responses of the parametric resonance of planar beam structures.

Experimental characterization of a nonlinear mechanical oscillator with softening behaviour for large displacements

This paper presents an experimental insight into the performance of a mechanical oscillator consisting of an X-shaped-spring configuration. This configuration achieves an overall softening characteristic with quasi-zero stiffness behaviour far away from the static equilibrium point. Such a geometrical nonlinear configuration has attracted significant research attention in the last few years, particularly for its application as a vibration isolator with the possibility to extend the quasi-zero-stiffness region beyond that of the classical three-spring nonlinear isolator. However, previous experimental evidence has been limited to small amplitude vibration excitation only. Furthermore, it has been focused mainly on the isolation region, rather than on the large amplitude response, thus circumventing an insight on the damping effects and its modelling. To address this gap, in this paper, both frequency sweeps and random excitations are applied to a prototype device for experimental characterization. A nonlinear stiffness model is developed based on the geometry of the system and a nonlinear damping model is assumed based on experimental observation. The proposed model accurately describes the dynamic behaviour of the system as shown by comparison of theoretical and experimental data.

Optimum stimulus for eliciting masseter vestibular-evoked myogenic potential: a comparative exploration with three different acoustic stimuli

ObjectiveTo compare the EMG rectified amplitude, absolute latencies, interpeak interval, and Interaural asymmetry parameters of masseter vestibular-evoked myogenic potential (mVEMP) elicited using clicks, 500 Hz tone bursts, and 500 Hz NB CE-chirps.MethodTwenty-five young healthy adults in the age range of 18–27 years participated for the study. mVEMP was recorded using three different acoustic stimuli i.e., clicks, 500 Hz tone bursts, and 500 Hz NB CE-chirps. mVEMP was recorded at an intensity of 125 dB peSPL with 5.1/s repetition rate. The potentials were recorded ipsilaterally using zygomatic electrode montage and were filtered between 0.1 and 3000 Hz. EMG rectification of the responses was made prior to analysis.ResultsThe latencies of P1 and N1 were significantly earlier for chirps then followed by click and tone bursts. The EMG rectified amplitude was significantly larger for the potentials obtained using chirps followed by tone bursts and then the clicks. Masseter VEMP obtained using chirps had significantly larger interpeak interval than tone bursts and clicks. The mean amplitude asymmetry ratio was greater in the potentials obtained using chirps than the other two stimuli.ConclusionThe present study reveals that 500 Hz NB CE-chirps tend to produce mVEMP with larger response amplitude and earlier latencies and thus are considered better and constructive stimuli compared to clicks and tone bursts.

Experimental Study on the Effect of the Angle of Attack on the Flow-Induced Vibration of a Harbor Seal’s Whisker

A harbor seal’s whisker is able to sense the trailing vortices of marine organisms due to its unique three-dimensional wavy shape, which suppresses the vibrations caused by its own vortex-shedding, while exciting large-amplitude and synchronized vibrations in a wake flow. This provides insight into the development of whisker-inspired sensors, which have broad applications in the fields of ocean exploration and marine surveys. However, the harbor seal’s whisker may lose its vibration suppression ability when the angle of attack (AoA) of the incoming flow is large. In order to explore the flow-induced vibration (FIV) features of a harbor seal’s whisker at various angles of attack (θ=0–90∘), this study experimentally investigates the effect of AoA on the vibration response of a whisker model in a wide range of reduced velocities (Ur = 3–32.2) and the Reynolds number, Re = 400–7000, in a circulating water flume. Meanwhile, for the sake of comparison, the FIV response of an elliptical cylinder with the same equivalent diameters is also presented. The results indicate that an increase in AoA enhances the vibration amplitude and expands the lock-in range for both the whisker model and the elliptical cylinder. The whisker model effectively suppresses vibration responses at θ=0∘ due to its unique three-dimensional wavy shape. However, when θ≥30∘, the wavy surface structure gradually loses its suppression ability, resulting in large-amplitude vibration responses similar to those of the elliptical cylinder. For θ = 30∘ and 45∘, the vibration responses of the whisker model and the elliptical cylinder undergo three vibration regimes, i.e., vortex-induced vibration, transition response, and turbulent-induced vibration, with the increasing Ur. However, at θ = 60∘ and 90∘, the vortex-shedding gradually controls the FIV response, and only the vortex-induced vibration is observed.

Next-Generation Auditory Steady-State Responses in Normal-Hearing Adults: A Pilot Test–Retest Reliability Study

The Auditory Steady-State Response (ASSR) provides objective and ear-specific information essential for early and appropriate intervention. Test–retest reliability is essential for audiological monitoring. The test–retest reliability of the ASSR has received limited attention. Only a handful of studies found in the literature investigated the test–retest reliability of old-generation ASSR using amplitude or mixed modulated stimuli. However, to our knowledge, no published reports have specifically examined the test–retest reliability of the next-generation ASSR using Chirp family stimuli as implemented in the Interacoustics Eclipse system. This pilot study investigated (a) the test–retest reliability of air conduction (AC) ASSR thresholds across two test sessions and (b) the relationship between differences in ASSR thresholds across two sessions to the residual noise levels in normal-hearing adults. Methods: Fifteen normal-hearing adults (12 females) (30 ears) with an average age of 28 years were recruited for the study. The ASSRs were recorded using a two-channel recording montage. The automatic default stimuli and recording protocol using 90 Hz ASSR, and the accuracy method (p < 0.01) as implemented in the Eclipse system is used to measure ASSR. Results: The study demonstrated strong test–retest reliability for ASSR across frequencies (500 Hz, 1000 Hz, 2000 Hz, and 4000 Hz). Notably, the highest reliability was observed at 500 Hz. The mean test–retest reliability of ASSR was found to be comparable to pure-tone thresholds, but the intra-subject variability is higher for ASSR compared to pure-tone thresholds. Additionally, no significant correlation was found between the difference in ASSR residual noise levels at the threshold and the difference in ASSR thresholds at all tested frequencies. Conclusion: The next-generation system demonstrated strong test–retest reliability across the frequencies examined in this pilot study. Particularly, an improvement in reliability was observed at 500 Hz compared to the old-generation ASSR. This enhancement can be attributed to the utilization of narrow-band CE-chirp stimuli, which generate large amplitude responses, and the implementation of an improved detection paradigm involving multiple harmonics spectral and phase analysis. This pilot study only enrolled adults with normal hearing, and future investigations should include a larger sample size comprising both normal-hearing and hearing-impaired individuals, as well as the pediatric population.

Dynamics of time-modulated, nonlinear phononic lattices.

The propagation of acoustic and elastic waves in time-varying, spatially homogeneous media can exhibit different phenomena when compared to traditional spatially varying, temporally homogeneous media. In the present work, the response of a one-dimensional phononic lattice with time-periodic elastic properties is studied with experimental, numerical and theoretical approaches in both linear and nonlinear regimes. The system consists of repelling magnetic masses with grounding stiffness controlled by electrical coils driven with electrical signals that vary periodically in time. For small-amplitude excitation, in agreement with linear theoretical predictions, wave-number band gaps emerge. The underlying instabilities associated to the wave-number band gaps are investigated with Floquet theory and the resulting parametric amplification is observed in both theory and experiments. In contrast to genuinely linear systems, large-amplitude responses are stabilized via the nonlinear nature of the magnetic interactions of the system, and results in a family of nonlinear time-periodic states. The bifurcation structure of the periodic states is studied. It is found the linear theory correctly predicts parameter values from which the time-periodic states bifurcate from the zero state. In the presence of an external drive, the parametric amplification induced by the wave-number band gap can lead to bounded and stable responses that are temporally quasiperiodic. Controlling the propagation of acoustic and elastic waves by balancing nonlinearity and external modulation offers a new dimension in the realization of advanced signal processing and telecommunication devices. For example, it could enable time-varying, cross-frequency operation, mode- and frequency-conversion, and signal-to-noise ratio enhancements.

Application of GDQ method to large amplitude response of FGM joined spherical-conical shells under rapid surface heating

Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) joined spherical-conical shells are analyzed in the current research. Thermo-mechanical properties of the shells are assumed to be temperature and position dependent. The system of joined spherical-conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and Crank-Nicolson methods. This equation is non-linear since the thermo-mechanical properties of the shells are temperature dependent. The total functional of the shells is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well-validated with the available data for the case of single FGM spherical shell. Parametric studies are conducted to examine the influences of conical and spherical shell geometries, material composition, temperature dependence, in-plane and out-of-plane mechanical boundary conditions, various configuration of conical/spherical shell system, and thermal boundary conditions. It is highlighted that thermally induced vibrations indeed exists.

Nonlinear finite element free and forced vibrations of cellular plates having lattice-type metamaterial cores: a strain gradient plate model approach

The main objective of this paper is to develop a theoretically and numerically reliable and efficient methodology based on combining a finite element method and a strain gradient shear deformation plate model accounting for the nonlinear free and forced vibrations of cellular plates having lattice-type metamaterial cores. The proposed model provides a modelling framework computationally more efficient than framework of the conventional three-dimensional elasticity for the simulation of the underlying nonlinear dynamics of cellular plates with microarchitectures. The corresponding governing equations follow Mindlin’s strain gradient elasticity theory including the micro-inertia effect applied to the first-order shear deformation plate theory along with the nonlinear von Kármán kinematics. Standard and higher-order computational homogenization methods determine the classical and strain gradient material constants, respectively. A higher-order C1-continuous 6-node finite element is adopted for the discretization in the spatial domain, and an arc-length continuation technique along with time-periodic discretization is implemented to solve the resulting nonlinear time-dependent problem. Through a set of comparative studies with 3D full-field models as reliable validation references for triangularly prismatic example cores, the accuracy and efficacy of the proposed dimension reduction methodology are demonstrated for a diverse range of problem parameters for analyzing the large-amplitude dynamic structural response. The results show that the proposed strain gradient plate model can accurately capture the microarchitecture effects of cellular plates with low computational costs, which signifies a relevant contribution particularly to the computational analysis of nonlinear dynamics.

A CFD Study of Vortex-Induced Motions of a Semi-Submersible Floating Offshore Wind Turbine

Vortex-induced motion (VIM) is a critical issue for floating structures made of one or more columns, due to its significant impacts on their operational stability. Supported by column-type floating platforms, floating offshore wind turbines (FOWTs) may also experience large-amplitude VIM responses in current flow. Existing research on FOWTs has mostly focused on their wind/wave induced responses, yet less attention has been paid to their responses in current flow. In this paper, the VIM of the OC4 semi-submersible FOWT platform is studied numerically over a wide range of flow velocity. Three incidence angles, i.e., 0°, 90°, and 180°, are considered and the effect of current incidence on platform VIM is analysed. Results show that the so-called lock-in phenomenon is present and that a large transverse response amplitude of more than 0.3D persists until Vr = 30, with its maximum reaching over 0.8D at Vr = 8. Meanwhile, the transverse response amplitude for cases with the incidence angle of 180° is generally smaller, with a narrower lock-in regime, than those under the other two incidence scenarios. Flow field visualisation reveals that upstream vortices continuously interact with the downstream side column when the incidence angle turns to 180°, impacting the vortex shedding process and consequently fluid forces of the downstream column.

Langmuir-Blodgett Films of Arachidic and Stearic Acids as Sensitive Coatings for Chloroform HF SAW Sensors.

Properties of the Langmuir-Blodgett (LB) films of arachidic and stearic acids, versus the amount of the films' monolayers were studied and applied for chloroform vapor detection with acoustoelectric high-frequency SAW sensors, based on an AT quartz two-port Rayleigh type SAW resonator (414 MHz) and ST-X quartz SAW delay line (157.5 MHz). Using both devices, it was confirmed that the film with 17 monolayers of stearic acid deposited on the surface of the SAW delay line at a surface pressure of 30 mN/m in the solid phase has the best sensitivity towards chloroform vapors, compared with the same films with other numbers of monolayers. For the SAW resonator sensing using slightly longer arachidic acid molecules, the optimum performance was reached with 17 LB film layers due to a sharper decrease in the Q-factor with mass loading. To understand the background of the result, Atomic Force Microscopy (AFM) in intermittent contact mode was used to study the morphology of the films, depending on the number of monolayers. The presence of the advanced morphology of the film surface with a maximal average roughness (9.3 nm) and surface area (29.7 µm2) was found only for 17-monolayer film. The effects of the chloroform vapors on the amplitude and the phase of the acoustic signal for both SAW devices at 20 °C were measured and compared with those for toluene and ethanol vapors; the largest responses were detected for chloroform vapor. For the film with an optimal number of monolayers, the largest amplitude response was measured for the resonator-based device. Conversely, the largest change in the acoustic phase produced by chloroform adsorption was measured for delay-line configuration. Finally, it was established that the gas responses for both devices coated with the LB films are completely restored 60 s after chamber cleaning with dry air.

Active Vibration Control of Nonlinear Elastic Wire Rope in Mine Hoisting System

Based on the importance of the safe operation of the mine hoist mechanism, an adaptive control method based on Fuzzy Sliding Mode Control (FSMC) was applied to synchronize the large-amplitude response of geometrically nonlinear elastic wire rope in mine hoist systems. To efficiently and precisely synchronize the lateral vibration of a wire rope under the adaptive synchronizing method, the control equation of the elastic wire rope was given considering the large von Karman geometrical deformation, and it was converted to a six-dimensional nonlinear dynamical system using third-order Galerkin discretization. Based on this dynamic system, the existing FSMC is improved, so that the improved control strategy will be available for the active dynamic control of multidimensional nonlinear dynamical systems. The research shows that the control strategy can effectively synchronize the large-amplitude response of wire rope, thus providing the basis for dynamic design, fault diagnosis, and safe operation of the mine hoist system.

Flapping dynamics of an inverted flag behind a cylinder

The inverted flag configuration is inspired by biological structures (e.g. leaves on a tree branch), showing rich dynamics associated with instabilities at lower flow speeds than the regular flag configuration. In the biological counterpart, the arrangement of leaves and twigs on foliage creates a complex interacting environment that promotes certain dynamic fluttering modes. While enabling a large amplitude response for reduced flow speeds is advantageous in emerging fields such as energy harvesting, still, little is known about the consequence of such interactions. In this work, we numerically study the canonical bio-inspired problem of the flow-structural interaction of a 2D inverted flag behind a cylindrical bluff body, mimicking a leaf behind a tree branch, to investigate its distinct fluttering regimes. The separation distance between the cylinder and flag is gradually modified to determine the effective distance beyond which small-amplitude or large-amplitude flapping occurs for different flow velocities. It is shown that the flag exhibits a periodic large amplitude−low frequency response mode when the cylinder is placed at a sufficiently large distance in front of the flag. At smaller distances, when the flag is within the immediate wake of the cylinder, the flag undergoes a high frequency−small amplitude response. Finally, the flag’s piezoelectric power harvesting capability is investigated numerically and experimentally for varying geometrical and electrical parameters associated with these two conditions. Two separate optimal response modes with the highest energy output have also been identified.

Nonlinear Reduced Order Modeling of Structures Near Buckling and Application to an Energy Harvester

Abstract The focus of this investigation is first on assessing the validity to structures under in-plane forces, in particular near buckling, of the reduced order modeling approach for nonlinear geometric response that has been extensively developed in the last two decades. This focus is motivated by a class of piezoelectric energy harvesters that rely on strongly nonlinear behavior, such as large amplitude responses, to achieve broadband energy harvesting. A simple, two-rigid bars linkage that approximates a buckling beam is first considered to discover the features of the nonlinear force–displacement relationship induced by an in-plane loading. It is observed that the corresponding form of this relationship is not consistent with the one derived from a reduced order model (ROM) but can be closely approximated by it over a large displacement range. This analysis emphasizes in particular the role of a group of ROM coefficients that are usually considered unimportant. A similar study is performed next for the buckled harvester modeled within nastran and it is again found that a close match of the force–displacement relationship can be achieved. Based on that positive outlook, a six basis functions ROM of this beam harvester that includes piezoelectric effects is built and identified. It is found to provide a close match of nastran nonlinear predictions over a broad range of transverse and in-plane loadings in static and dynamic conditions. The ROM usefulness in predicting the open-circuit voltage is demonstrated.

Numerical investigation on the effect of diameter ratio on wake induced vibrations of tandem cylinders

Numerical simulations of WIV (wake induced vibration) for tandem cylinders with different diameter ratio in the impingement regime are performed for a low Reynolds number of Re = 150 and low mass ratio of m* = 2.0. Five different diameter ratios of d/D = 1/2, 3/4, 1, 4/3, and 3/2 are examined, in order to identify the effects of diameter ratio on 2-DOF (two-degree-of-freedom) vibration responses of the cylinder in the range of 2≤Ur ≤ 14. The results show that the diameter ratio and vortex mode play important roles in the vibration amplitudes and frequencies of the downstream cylinder. The WIV can be further classified into WIEV (wake interference and enhancement vibration) and WCV (wake controlled vibration) according to two distinct mechanisms of the amplification of cross-flow vibration amplitude. As d/D ≤ 1, the vibration frequencies of the downstream cylinder are significantly pulled down and interfered by the small vortices shedding from the small upstream cylinder. Meanwhile, a wider range of lock-in and enhanced vibration amplitudes are the features for WIEV. The inline direction of the downstream cylinder have large response amplitudes in a wide range of Ur (reduced velocity) and reach to the maximum value at d/D = 1. As d/D > 1, the hydrodynamic characteristics of the downstream cylinder are mainly controlled by the larger vortices shedding from the upstream cylinder. The downstream cylinder vibrates as the vortex shedding frequency of the larger upstream cylinder and the lock-in phenomenon is absent. WCV significantly increases the risk of fatigue failure of the cylindrical structure.

Flow-induced vibration of flexible cylinders covered by fixed fairings with different chord-thickness ratios

Fairings are important vortex-induced vibration (VIV) suppression devices in the offshore and marine engineering applications. For circular cylinders in the marine structures, both their mean deflection in the in-line direction and the oscillation amplitude in the cross-flow direction can be effectively reduced when the fairings work properly. However, plenty recent investigations report that when the rotational elements of fairings fail to function, the VIV suppression functions may be deprived and severe galloping oscillations may occur. However, very few studies discussed the galloping oscillation mechanism for slender flexible cylinders. We perform a numerical study on the fluid–structure interaction of flexible cylinders covered by fixed fairings with different chord-thickness ratios (c/t) using a self-developed fluid–structure interaction simulation code. The Reynolds number is 1.68×104, which is based on the incoming velocity and the diameter of the cylinder. It is found that large amplitude, low frequency and low order galloping responses are prominent in the cases where the flexible cylinder is covered by fairings with large c/t. The large added mass and the low fluid force oscillation frequency in galloping are found to induce the low vibration frequency and the low order ‘lock-in’ modes in flexible cylinder responses.

Vibration Characteristics and Experimental Research of Combined Beam Tri-Stable Piezoelectric Energy Harvester.

Vibration energy harvesting technology is expected to solve the power supply and endurance problems of wireless sensor systems, realize the self-power supply of wireless sensor systems in coal mines, and promote the intelligent development of coal mine equipment. A combined beam tri-stable piezoelectric energy harvester (CTPEH) is designed by introducing magnetic force into the combined beam structure. In order to explore the vibration characteristics of CTPEH, a nonlinear magnetic model is obtained based on the magnetic dipole theory, and the dynamic equation of the system is established using the Lagrange theorem and Rayleigh–Ritz theory. The influence of the different magnet distances and excitation conditions on the static bifurcation characteristics and dynamic response characteristics of the system are analyzed by numerical simulation, and the simulation results are validated by the experiments. The research results show that the motion state of the CTPEH system has four transition forms from mono-stable to tri-stable with the change in magnet distance. The tri-stable system has three potential energy curves with different characteristic shapes. The appropriate starting excitation position and excitation frequency can make it easier for the system to realize a large-amplitude response state, thereby improving the output performance of the system. This research provides new ideas and methods for optimizing the performance of the combined beam piezoelectric energy harvester.

Multi-Mode Motion Dynamics of a Top Tensioned Riser Excited by Vortices and Varying Tension

A coupled model of a top tensioned riser (TTR) under combined excitations from vortices and time-varying tension is proposed. And the van der Pol oscillator is introduced to simulate the time-varying characteristics of vortex shedding. The first three mode approximations as a six degree-of-freedom (DOF) system of vibrations of the fluid-structure interaction system are derived. Two cases of resonances are analyzed by using the numerical method. One case is on the vortex induced vibration (VIV) of the six DOF system, where the 1-to-1 internal resonance (also named “lock-in” phenomenon) between frequencies of the first modes of structure and the wake oscillator are investigated. Bifurcation curves between dynamic responses in terms of the shedding frequency are derived. Comparing to the vibrations of the first order mode, the “lock-in” phenomenon for vibrations of the higher order modes can occur in the wider parameter range with large values. The second case is on the combined resonance between the 1-to-1 internal and 1-to-2 parametric resonance. Response curves versus the shedding frequency under different amplitude ratios are calculated by the Runge–Kutta method. Effects of system parameters including the amplitude ratio of the parametric excitation, shedding frequency on the response curves are investigated. Comparing to the case when only considering the 1-to-1 internal resonance in the VIV, results of the combined resonance show those large-amplitude responses of the structure and wake oscillator can be induced. Responses will increase rapidly as the amplitude of the varying tension increases. Unstable responses can occur within a large frequency band of nondimensional shedding frequency. Furthermore, the dynamic characteristics of the responses of the six DOF system including the bifurcation diagrams, time histories, phase-plane projection and Poincaré sections are computed for different amplitude ratios. Results indicate that complex dynamics such as the quasi-periodic and chaotic solutions can occur for different amplitude ratios. The analysis can be helpful for the parameter design of deep-water risers.