Static Bifurcation Analysis Research Articles

Bio-inspired quad-stable piezoelectric energy harvester for low-frequency vibration scavenging

Due to the unique and diverse nonlinear characteristics, bionic structures can effectively broaden the acquisition bandwidth and improve the performance of harvesters. Inspired by dipteran flight motion, a novel bionic quad-stable piezoelectric energy harvester (BQPEH) is designed to collect low-frequency ambient vibration. The novelty of this bionic energy harvester lies in the combination of the “click” mechanism and “snap-through” motion, thus broadening the application scenarios and improving adaptability for the harvesters. A mass block and a post-buckled piezoelectric beam are connected by a rigid bar to create quadruple-well potentials. The static bifurcation analysis is performed to identify the quad-stable region in the parameter space. The dynamic responses of the system are investigated theoretically and numerically under constant and swept-frequency excitations and then verified experimentally. Broadband low-frequency and high-power energy harvesting are obtained through the inter-well motion induced by the combined nonlinearity of the BQPEH. The results indicate the potential of the bionic strategy for powering wireless devices in low-frequency environments and provide a novel concept for the innovative development of energy harvesters.

Design and investigation of a quad-stable piezoelectric vibration energy harvester by using geometric nonlinearity of springs

In this paper, a quad-stable piezoelectric vibration energy harvester induced by the geometric nonlinearity of springs is designed and investigated. The novelty is that by replacing the permanent magnets with a combination of springs, the system can be effectively protected from electromagnetic interference in the working environment. The innovative design of three hinged springs attached to the tip of the cantilever beam makes the stiffness of the structure inherently nonlinear, which can induce the multi-stability of the system. The Euler-Lagrange equations are applied to establish the governing equation of the energy harvester. The static bifurcation analysis reveals the evolution process of multi-stability. The complex dynamic frequency (CDF) method, an efficient approach to solving strongly nonlinear dynamic problems with multi-well systems, is then applied to deduce the system's amplitude-frequency response. Combinations of theoretical and numerical results show that among a wide frequency bandwidth, the system can exhibit such as single-, double- and quadruple-well periodic motions. Finally, experiments are set up to verify the advantage of inter-well motion on harvesting power. Result shows that geometric nonlinearity can effectively make the energy harvester exhibit multi-stability, generate different movement patterns, and obtain high harvesting power.

Dynamic Design of a Quad-Stable Piezoelectric Energy Harvester via Bifurcation Theory

The parameter tuning of a multi-stable energy harvester is crucial to enhancing harvesting efficiency. In this paper, the bifurcation theory is applied to quantitatively reveal the effects of structural parameters on the statics and dynamics of a quad-stable energy harvester (QEH). Firstly, a novel QEH system utilizing the geometric nonlinearity of springs is proposed. Static bifurcation analysis is carried out to design quad-stable working conditions. To investigate the cross-well and high-energy vibration, the complex dynamic frequency (CDF) method, suitable for both weakly and strongly nonlinear dynamic problems, is then applied to deduce the primary response solution. By using the unfolding analysis in singularity theory, four steady-state properties and dozens of primary resonance modes are demonstrated. Based on the transition set, the effective bandwidth for energy harvesting can be customized to adapt well to various vibration environments by parametric adjustment. Finally, the experimental tests verify that the output power can reach up to 1 mW. The proposed QEH and its mechanics optimization can guide energy supply for next-generation wireless systems and low-power sensors under magnetic forbidding environments.

Enhancing the Recognition Task Performance of MEMS Resonator-Based Reservoir Computing System via Nonlinearity Tuning.

Reservoir computing (RC) is a potential neuromorphic paradigm for physically realizing artificial intelligence systems in the Internet of Things society, owing to its well-known low training cost and compatibility with nonlinear devices. Micro-electro-mechanical system (MEMS) resonators exhibiting rich nonlinear dynamics and fading behaviors are promising candidates for high-performance hardware RC. Previously, we presented a non-delay-based RC using one single micromechanical resonator with hybrid nonlinear dynamics. Here, we innovatively introduce a nonlinear tuning strategy to analyze the computing properties (the processing speed and recognition accuracy) of the presented RC. Meanwhile, we numerically and experimentally analyze the influence of the hybrid nonlinear dynamics using the image classification task. Specifically, we study the transient nonlinear saturation phenomenon by fitting quality factors under different vacuums, as well as searching the optimal operating point (the edge of chaos) by the static bifurcation analysis and dynamic vibration numerical models of the Duffing nonlinearity. Our results in the optimal operation conditions experimentally achieved a high classification accuracy of (93 ± 1)% and several times faster than previous work on the handwritten digits recognition benchmark, profit from the perfect high signal-to-noise ratios (quality factor) and the nonlinearity of the dynamical variables.

Influence of Controller’s Parameters on Static Bifurcation of Magnetic-Liquid Double Suspension Bearing

Magnetic-Liquid Double Suspension Bearing (MLDSB) is composed of an electromagnetic supporting and a hydrostatic supporting system. Due to greater supporting capacity and static stiffness, it is appropriate for occasions of middle speed, overloading, and frequent starting. Because of the complicated structure of the supporting system, the probability and degree of static bifurcation of MLDSB can be increased by the coupling of hydrostatic force and electromagnetism force, and then the supporting capacity and operation stability are reduced. As the key part of MLDSB, the controller makes an important impact on its supporting capacity, operation stability, and reliability. Firstly, the mathematical model of MLDSB is established in the paper. Secondly, the static bifurcation point of MLDSB is determined, and the influence of parameters of the controller on singular point characteristics is analyzed. Finally, the influence of parameters of the controller on phase trajectories and basin of attraction is analyzed. The result showed that the pitchfork bifurcation will occur as proportional feedback coefficient Kp increases, and the static bifurcation point is Kp = −60.55. When Kp < −60.55, the supporting system only has one stable node (0, 0). When Kp > −60.55, the supporting system has one unstable saddle (0, 0) and two stable non-null focuses or nodes. The shape of the basin of attraction changed greatly as Kp increases from −60.55 to 30, while the outline of the basin of attraction is basically fixed as Kp increases from 30 to 80. Differential feedback coefficient Kd has no effect on the static bifurcation of MLDSB. The rotor phase trajectory obtained from theoretical simulation and experimental tests are basically consistent, and the error is due to the leakage and damping effect of the hydrostatic system within the allowable range of the engineering. The research in the paper can provide theoretical reference for static bifurcation analysis of MLDSB.

A low-frequency, wideband quad-stable energy harvester using combined nonlinearity and frequency up-conversion by cantilever-surface contact

Abstract This paper proposes models and experiments of a wideband piezoelectric vibration energy harvester with a quadruple-well potential induced by the combined nonlinearity of cantilever-surface contact and magnetoelasticity. The cantilever-surface contact introduces frequency up-conversion mechanism into the present energy harvester and provides favorable mechanical properties such as minimal moving parts, low frictional losses and high shock resistance. The magnetoelasticity, combined with the cantilever-surface contact, can make the present energy harvester exhibit multi-stability, ranging from mono-stability to quad-stability. The formation mechanism of this multi-stability is investigated by a static bifurcation analysis to the harvester’s mathematical model. Dynamic responses of the present energy harvester with a quadruple-well potential are explored by numerical simulations and validated by experiments. The results show that the combined nonlinearity can not only improve the efficiency of electrical power transfer under low intensity excitations, but also extend the wide operating band of harvester to low frequencies.

Nonlinear dynamic analysis of electrically actuated viscoelastic bistable microbeam system

Viscoelastic phenomena widely exist in MEMS materials, which may have certain effects on transition mechanism of nonlinear jumping phenomena and transient chaotic behaviors. This article aims to theoretically investigate the static and dynamic characteristics of electrically actuated viscoelastic bistable microbeam via a low-dimensional model. An improved single-degree-of-freedom model to describe microbeam-based resonators is obtained by using Fractional Kelvin constitutive model, Hamilton’s principle and Galerkin method. Through static bifurcation analysis, three kinds of parameter conditions of the bistable system are obtained, and potential energy function of the Hamiltonian system is theoretically derived. The influence of fractional viscoelasticity on dynamic pull-in phenomena is distinguished from the viewpoint of energy. Then, the method of multiple scales is applied to determine the response and stability of the system for small vibration amplitude and AC voltage. The influence of fractional viscoelasticity on amplitude, frequency and bifurcation behavior is investigated. Results show that compared with the elastic material, nonlinear phenomenon becomes weak, resonance frequency increases and amplitude decreases in the viscoelastic system. Besides, the numerical discretization method of fractional derivative is given to verify theoretical results. To study the influence of fractional viscoelasticity on complicated vibration, Melnikov method is applied to predict the existence of chaos, and numerical simulation is carried out to find the stable regions, chaotic regions and dynamic pull-in regions by using bifurcation diagrams with local maximum method. Rational increase in material modulus ratio parameter and fractional order is effective to reduce the possibility of chaos and dynamic pull-in. This analysis has the potential of developing parameter design in MEMS.

Static bifurcation analysis of natural circulation inner evaporative cooling system in wind turbine

Wind power is one of the most attractive renewable clean energies under development at present. On a global scale, wind power generation development was very rapid in recent years. As the wind power generation tends to develop toward large-scale and offshore, the traditional cooling methods gradually expose their own shortcomings. As the large wind turbine installation tower is high, and the installation site is dispersed, the installation and maintenance of wind turbine generator are difficult. So, the generator is required to have a small weight and less maintenance. Self-circulation inner evaporative cooling system (SCIECS) has the following advantages: self-circulation without pump, high cooling efficiency, safe and reliable operation, and basically maintain-free, etc. The self-circulation of cooling system can be realized by the 35 between wind turbine generator and the horizontal direction, and it is very suitable for being used in a large-scale wind power generator. Owing to the intrinsic nonlinearity of two-phase self-circulation system, changes of operation condition and circuit topology in a large range may lead to an instability of the cooling system, causing the system parameters to severly change. The instability of the cooling system can cause the local overheating and even burning of the generator, which provides a huge security risk for the cooling system, thus threatening the safe and stable operation of the generator. The stability of SCIECS is very important for the safe operation of wind turbine. In this paper, static stability of the SCIECS in wind power generator is studied based on the nonlinear bifurcation analysis theory and its numerical continuation method. System static bifurcation diagram is obtained to analyze the evolution characteristics of the SCIECS. Parameter effect of the system static bifurcation is analyzed at the same time. In order to verify the theoretical prediction of the static bifurcation of the small-angle two-phase natural circulation, an experimental platform is built. Static bifurcation of the SCIECS is observed experimentally. The experimental results show that the static bifurcation phenomenon exists in the natural circulation two-phase flow of small angle, and the theoretically predicted m-Q bifurcation curves are in good agreement with the experimental curves, which verifies the correctness of the theoretical analysis.

Static bifurcation and primary resonance analysis of a MEMS resonator actuated by two symmetrical electrodes

This paper investigates the static and dynamic characteristics of a doubly clamped micro-beam-based resonator driven by two electrodes. The governing equation of motion is introduced here, which is essentially nonlinear due to its cubic stiffness and electrostatic force. In order to have a deep insight into the system, static bifurcation analysis of the Hamiltonian system is first carried out to obtain the bifurcation sets and phase portraits. Static and dynamic pull-in phenomena are distinguished from the viewpoint of energy. What follows the method of multiple scales is applied to determine the response and stability of the system for small vibration amplitude and AC voltage. Two important working conditions, where the origin of the system is a stable center or an unstable saddle point, are considered, respectively, for nonlinear dynamic analysis. Results show that the resonator can exhibit hardening-type or softening-type behavior in the neighborhood of different equilibrium positions. Besides, an attractive linear-like state may also exist under certain system parameters if the resonator vibrates around its stable origin. Whereafter, the corresponding parameter relationships are deduced and then numerically verified. Moreover, the variation of the equivalent natural frequency is analyzed as well. It is found that the later working condition may increase the equivalent natural frequency of the resonator. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Application of Bifurcation Theory to Voltage Stability in Power System

In order to study on the problem of voltage stability of power system, this paper describes the static bifurcation analysis and the dynamic bifurcation analysis in voltage stabilization analysis of power system and its relationship with the voltage stability,discusses the voltage instability caused by two main bifurcation formal definition, the occurrence of the conditions and the calculation of the bifurcation point, and points out advantages and disadvantages of various algorithms. Finally the paper looks forward to further study of the bifurcation theory in terms of voltage stability.

Non-linear dynamic behaviour of multi-folding microstructure systems based on origami skill

This paper presents the theoretical basis for both static and dynamic numerical approaches to the elastic stability and elasto-plastic stability of a folding multi-layered truss. Both analyses are based on bifurcation theory and include geometrical non-linearity. The dynamic analysis includes an allowance for contact between nodes. Comparisons are made between elastic behaviour and elasto-plastic behaviour obtained from both numerical dynamic methods in which bifurcations are demonstrated as a theoretically elastic analysis. Our analysis reveals that several folding behaviours in the pantographic truss are identified during the elastic and elasto-plastic instabilities, where the folding behaviour of the truss is shown to be a function of the initial geometry and velocity of the dynamic loading. The authors suggest that understanding this behaviour will be very useful for developing light-weight structures subject to dynamic loading based on static bifurcation analysis and dynamic analysis.

Perturbation Methods for Bifurcation Analysis from Multiple Nonresonant Complex Eigenvalues

It is shown that the logical bases of the static perturbation method, which is currently used in static bifurcation analysis, can also be applied to dynamic bifurcations. A two-time version of the Lindstedt–Poincare Method and the Multiple Scale Method are employed to analyze a bifurcation problem of codimension two. It is found that the Multiple Scale Method furnishes, in a straightforward way, amplitude modulation equations equal to normal form equations available in literature. With a remarkable computational improvement, the description of the central manifold is avoided. The Lindstedt–Poincare Method can also be employed if only steady-state solutions have to be determined. An application is illustrated for a mechanical system subjected to aerodynamic excitation.

Nonlinear kink modes for supersonic vortex sheets

Both recent large-scale numerical simulations and time-dependent asymptotic nonlinear wave theories reveal the prominence of kink modes in the nonlinear instability of supersonic vortex sheets. These kink modes are nonlinear traveling waves that move along the vortex sheet at various speeds and have a wave structure consisting of a kink in the slip stream bracketed by shocks and rarefactions emanating from each side of the kink. Here an explicit construction is developed for calculating all the nonlinear kink modes that bifurcate from a given unperturbed contact discontinuity. This construction is applied at small amplitudes to provide a completely independent confirmation of the asymptotic nonlinear wave theories through a static bifurcation analysis. For the unperturbed vortex sheet, bifurcation diagrams at large amplitudes are also computed for several interesting density ratios and Mach numbers. These results are applied at large amplitude to explain some of the phenomena observed in numerical simulations.

On the steady state solutions of chemically reacting systems-I. Applications to single species surface reactions

The application of singularity theory to the static bifurcation analysis of single species gas-solid catalytic reaction systems is shown to determine completely their steady state behaviour. The reactor residence time, inlet pressure of the reactant and catalyst surface area are used bifurcation parameters. Adsorbate-adsorbate interactions are quantified by a linear increase of the activation energy for desorption with coverage. The surface reaction is assumed to require either a single site or an additional empty site. In the former case, up to three steady states are possible, whereas in the latter, five steady states can appear for certain values of the parameters. The singularity theory analysis of these single species surface reactions shows new cases of qualitative behaviour, when the system is confined on the boundaries of the physical domain. Finally, the results are discussed within the context of realistic values of model parameters.