Bifurcation Topology Research Articles

Experimental Investigation of Mode Localization's Bifurcation Topology Transfer in Electrostatically Coupled Tuning Fork Resonators.

Bifurcation topology transfer phenomena in the presence of mode localization are investigated using double-ended fixed electrostatically coupled tuning fork resonators. An analytical model is proposed for the coupled tuning fork resonators, and the effects of feedthrough capacitance on the structure are also analyzed and eliminated by means of data post-processing. Then, an open-loop experimental platform is established, when the system is in balance state, the quality factor is obtained under test as Q = 9858, and comparison of the experiment with numerical simulation results is in good agreement. Finally, with the voltage increases, the two resonators gradually exhibit nonlinear characteristics. It is worth noting that when one of the coupled resonators exhibits nonlinear vibration behavior, even though the vibration amplitude of the other resonator is lower than the critical amplitude, it still exhibits nonlinear behavior, and the results confirm the existence of the bifurcation topology transfer phenomenon in coupled resonators' mode localization phenomenon.

Auto-parametric resonance of flexible viscoelastic beams under interaction between longitudinal and transverse modes

For a system encompassing subsystems, dynamic modal interaction may experience unanticipated vibrations among subsystems that are excited indirectly. This paper focuses attention to a nonlinear auto-parametric resonance phenomenon of a flexible beam introduced via modal coupling of longitudinal forced resonance and transverse parametric resonance under longitudinal pulsatile excitation. The coupled resonance equations are first modeled based on the Euler-Bernoulli beam theory by incorporating nonlinear curvature, von Kármán's nonlinear strain-displacement relationship and the flexible beam model characterizing nonlinear inertia and displacement interaction. The Galerkin truncation-incremental harmonic balance method (IHBM) is then introduced to obtain both the stability boundary and amplitude-frequency bifurcation maps. Through the corresponding relationship between stability boundary and bifurcation diagram, the energy exchange between two modes is identified as well as the instability region and bifurcation topology can be tuned. It is found that the modal interaction greatly widens the instability region with at most two resonance bands of lateral resonance. By tuning structural and foundation damping, resonance can be controlled to occur only in one frequency band of the pair as desired. This research provides guidance to design broadband energy harvesting or vibration suppression equipment with tunable working frequency band for using the strategy of auto-parametric resonance.

Parametric-forced coupling resonance of core-shell nanowires with interfacial damage under weak viscoelastic boundary constraint

Revealing the synergistic effects of size-dependency and nonlinearity on dynamics of core-shell nanowires plays an important role in nonlinear design of nanoelectromechanical system (NEMS) maintaining stable working state. The nonlinear principal resonance of core-shell nanowires with weak interface under parametric-forced coupling excitations is studied. The nanowire is simply supported at both ends, with one end movable and under a weak viscoelastic boundary described by Kelvin-Voight model. The weak interface obeys the interfacial cohesive law coupled into the refined displacement field. Geometric nonlinearity due to large deformation is reflected in three aspects. Stiffness nonlinearity is modeled by nonlinear curvature. Longitudinal inertia nonlinearity is modeled based on an in-extensive beam assumption. Nonlinear constraint force provided by the viscoelastic boundary is reflected in nonlinear damping and stiffness. Size-dependency including nonlocal stress, strain gradient and surface elasticity are all incorporated. The governing equations are derived by Hamilton's variational principle and solved via a perturbation-incremental harmonic balance method (IHBM). Results provide the bifurcation diagram of coupled resonance and the stability boundary of parametric resonance for analyzing the change mechanism of bifurcation topology. Decisive ways are revealed that weak interface, size dependency and constraints from foundation and boundary affect the linear and nonlinear properties of the system.

Coupled resonance of FGM nanotubes transporting super-critical high-speed pulsatile flow under forced vibration: size-dependence and bifurcation topology

The nano electromechanical system (NEMS) with functionally graded (FG) nanotubes as the core component is usually in the multi field environment leading to multi-source excitation resonance. As a typical and basic excitation form, the nonlinear principal resonance of FG nanotubes under the supercritical pulsatile flow and forced excitation is studied. Geometric nonlinearity originates from the tensile hardening of immovable boundary modeled​ by von Kármán’s nonlinear geometric relationship. Size-dependencies for both nanofluids and nanosolids are characterized by slip-flow model and nonlocal strain gradient model coupled with surface effects, respectively. Under the framework of Zhang–Fu’s beam theory, the governing equations for coupled resonance based on the initial post-buckling equilibrium state are established. A combination of perturbation-incremental harmonic balance method (IHBM) is established and employed in bifurcation analysis of such a strongly nonlinear coupled resonance system for the first time The softening-type nonlinear behavior and multiple-stable solutions are found and two dominant factors controlling the bifurcation topology are revealed. It is suggested that the size effects of both nanoflow and nanosolid are dual and may change the bifurcation topology through two mechanisms. One is the change of two excitations from strong interaction to weak interaction, and the other is the change of multi-solution threshold of the two-excitation amplitude ratio. The results can guide the nonlinear design and adjustment of the NEMS to obtain available working frequency band maintaining stable motion state.

Electrostatic nonlinear dispersive parametric mode interaction

Understanding and controlling the nonlinear coupling in micro/nanomechanical resonators are of great importance to the exploitation of advanced devices. The recently observed electrostatic nonlinear parametric coupling is a very interesting topic. However, the theoretical model of the electrostatic parametric coupling remains unclear. This paper explicitly derives the model and the electrostatically induced dispersive parametric coupling which reveals the ability to tune the bifurcation topology of capacitive resonators is analyzed based on the multiple-time-scale method. A novel displacement-to-frequency transduction scheme based on this electrostatic dispersive parametric coupling effect is proposed. The transduction sensitivity is theoretically given, which indicates that this electrostatic dispersive transduction scheme can provide even more design freedoms than the existing displacement-to-frequency transduction scheme based on tension modulation. In addition, a bifurcation reversal effect is predicted in the strong actuated states of the dispersive parametric coupled system, which reveals the ability to tune the bifurcation topology of capacitive resonators.

Dynamic instability mechanism of post-buckled FG nanotubes transporting pulsatile flow: size-dependence and local/global dynamics

For bistable structures such as post-bucked nanotubes, the size-dependence may have influences on both local and global dynamics. A local/global dynamic instability study of post-buckled nanotubes transporting pulsatile flow is conducted for the first time. To model the size-dependence of both the solid and fluid, the nonlocal strain gradient beam model coupled with surface elasticity is established, and the slip-flow model is introduced. The governing equations are derived based on Zhang-Fu's refined beam theory and von Kármán type geometric relationship. A new computation mode is developed that combines the two-step perturbation- Galerkin truncation, incremental harmonic balance method (IHBM), and Runge-Kutta method to obtain the stability boundary and amplitude-frequency bifurcation diagram. Also, the dynamic characteristics in different parameter regions are presented through displacement-time history diagram, phase diagram, Poincaré map and spectrum diagram. The complex motions of local attractor- two-well chaos- global snap-through, and hysteresis are observed. Subsequently, the nonlinear dynamic instability mechanism and the physical significance of stability boundary are revealed. It is revealed that the structural size-dependence have dual effects and may change the bifurcation topology, such as the change of motion from local attractor to global attractor, and from periodic motion to chaotic motion.

Study on a Class of Piecewise Nonlinear Systems with Fractional Delay

In this paper, a dynamic model of piecewise nonlinear system with fractional-order time delay is simplified. The amplitude frequency response equation of the dynamic model of piecewise nonlinear system with fractional-order time delay under periodic excitation is obtained by using the average method. It is found that the amplitude of the system changes when the external excitation frequency changes. At the same time, the amplitude frequency response characteristics of the system under different time delay parameters, different fractional-order parameters, and coefficient are studied. By analyzing the amplitude frequency response characteristics, the influence of time delay and fractional-order parameters on the stability of the system is analyzed in this paper, and the bifurcation equations of the system are studied by using the theory of continuity. The transition sets under different piecewise states and the constrained bifurcation behaviors under the corresponding unfolding parameters are obtained. The variation of the bifurcation topology of the system with the change of system parameters is given.

A structured study on the dynamic bifurcation behavior of a continuous ethanol fermentor

The oscillatory behavior of a continuous ethanol fermentor, in the high feed sugar concentration region (200 g/L), is analyzed using the bifurcation theory. The mathematical model was solved using the arc-length continuation method including a refinement algorithm based on the Heaviside step function technique. Experimental data for ethanol production using Zymomonas mobillis were fitted to a phenomenological model based on the concept of dynamic specific growth rate and inhibition. Thus, a through connection is defined between model parameters and oscillatory behavior. Our approach shows the rich static and dynamic bifurcation topology of the continuous fermentation reactor, which includes Fold, Hopf, Bautin, and Cusp bifurcations. The four-dimensional model shows that intricate oscillations can occur and that the periodic attractors of the system can have thoughtful consequences on reactor performance. In fact, it is evidenced that, in some conditions, periodic/bifurcation behavior presents higher ethanol productivity than the corresponding steady-state. Bifurcation analysis provides insight into the possible operation conditions and their impact on conversion and productivity of the fermentation process.

Experimental characterization of nonlinear static and dynamic behaviors of circular capacitive microplates with initial deflection

In this paper, the effects of initial deflection on the static and dynamic behaviors of circular capacitive transducers are experimentally investigated. The obtained results are in good agreement with numerical simulations. It is shown that the initial deflection has a major impact on the static response of the resonator by shifting the pull-in voltage, and on its dynamic response by increasing the resonance frequency and modifying the bifurcation topology from softening to hardening behavior. Moreover, the dynamic behavior of the microplate may display nonlinear periodic and quasiperiodic responses due to geometric and electrostatic nonlinearities.

Subharmonic resonance bifurcation and chaos of simple pendulum system with vertical excitation and horizontal constraint

In order to improve the working performance and optimize the working parameters of the typical engineering pendulum of a typical system that it is abstracted as a physical simple pendulum model with vertical excitation and horizontal constraint. The dynamical equation of the system with vertical excitation and horizontal constraint is established by using Lagrange equation. The multiple-scale method is used to analyze the subharmonic response characteristics of the system. The amplitude-frequency response equation and the phase-frequency response equation are obtained through calculation. The effects of the system parameters on the amplitude resonance bandwidth and variability are clarified. According to the singularity theory and the universal unfolding theory, the bifurcation topology structure of the subharmonic resonance of the system is obtained. The Melnikov function is applied to the study of the critical conditions for the chaotic motion of the system. The parameter equation of homoclinic orbit motion is obtained through calculation. The threshold conditions of chaos in the sense of Smale are analyzed by solving the Melnikov function of the homoclinic motion orbit. The dynamic characteristics of the system, including single-parameter bifurcation, maximum Lyapunov exponent, bi-parameter bifurcation, and manifold transition in the attraction basin, are analyzed numerically. The results show that the main path of the system entering into the chaos is an almost period doubling bifurcation. Complex dynamical behaviors such as periodic motion, period doubling bifurcation and chaos are found. The bi-parameter matching areas of the subharmonic resonance bifurcation and chaos of the system are clarified. The results reveal the global characteristics of the system with vertical excitation and horizontal constraint, such as subharmonic resonance bifurcation, periodic attractor multiplication, and the coexistence of periodic and chaotic attractors. The results further clarify the mechanism of the influence of system parameters change on the movement form transformation, energy distribution and evolution law of the system. The mechanism of the influence of relevant parameters on the performance of the engineering system with vertical excitation and horizontal constraint is also obtained. The results of this research provide theoretical bases for adjusting the parameters of working performances of this typical physical system in engineering domain and the vibration reduction and suppression of the system in actual working conditions.

Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers.

In order to investigate the effects of geometric imperfections on the static and dynamic behavior of capacitive micomachined ultrasonic transducers (CMUTs), the governing equations of motion of a circular microplate with initial defection have been derived using the von Kármán plate theory while taking into account the mechanical and electrostatic nonlinearities. The partial differential equations are discretized using the differential quadrature method (DQM) and the resulting coupled nonlinear ordinary differential equations (ODEs) are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). It is shown that the initial deflection has an impact on the static behavior of the CMUT by increasing its pull-in voltage up to 45%. Moreover, the dynamic behavior is affected by the initial deflection, enabling an increase in the resonance frequencies and the bistability domain and leading to a change of the frequency response from softening to hardening. This model allows MEMS designers to predict the nonlinear behavior of imperfect CMUT and tune its bifurcation topology in order to enhance its performances in terms of bandwidth and generated acoustic power while driving the microplate up to 80% beyond its critical amplitude.

Internal resonances and their bifurcations of a rigid-flexible space antenna

The nonlinear dynamics of a flexible space antenna supported by a serial of two rigid arms mounted on a heavy satellite is studied under the conditions of 2:1 or 3:1 internal resonances. The nonlinear dynamic equations of the in-plane vibration of the antenna system of three degrees of freedom are derived first via the method of assumed modes. Then, the resonance and non-resonance regions are determined in terms of the dimensionless parameters of a rigid arm and a torsional spring in the antenna system. Afterwards, the approximate analytic solutions of frequency-amplitude response of the antenna system under 2:1 or 3:1 internal resonances are derived by using the method of multiple scales. Furthermore, the topologic description for the bifurcation of each internal resonance with respect to unfolding parameters is studied by means of the theory of singularity. Finally, the accuracy of the approximate analytic solutions is validated via the numerical solutions. The study shows that the internal resonances exist over a wide range of the detuning parameter. In addition, the 3:1 internal resonance exhibits much more complicated bifurcation topology than the 2:1 internal resonance. The approximate analytic solutions of internal resonances and their bifurcation analysis results provide an essential reference for designing the antenna system of concern.

Investigation of modal interactions and their effects on the nonlinear dynamics of a periodic coupled pendulums chain

The nonlinear dynamics of a weakly coupled pendulums chain is investigated under primary resonance. The coupled equations governing the nonlinear vibrations are normalized and transformed into a set of coupled complex algebraic equations using the multiple scales method coupled with standing wave decomposition. A model reduction method is proposed to calculate the dominant dynamics without significant loss of accuracy compared to the full model. The validity of the proposed semi-analytical method is verified, and its role in identifying the type of the solution branches is highlighted. The modal interactions and their effects on the nonlinear dynamics are studied in the frequency domain in order to emphasize the large number of multimode solution branches and the bifurcation topology transfer between the modal intensities. Basins of attraction analysis have been performed, showing that the distribution of the multimodal solution branches generated by all modes collectively increases by increasing the number of coupled pendulums.

Pattern selection and self-organization induced by random boundary initial values in a neuronal network

Abstract Regular spatial patterns could be observed in spatiotemporal systems far from equilibrium states. Artificial networks with different topologies are often designed to reproduce the collective behaviors of nodes (or neurons) which the local kinetics of node is described by kinds of oscillator models. It is believed that the self-organization of network much depends on the bifurcation parameters and topology connection type. Indeed, the boundary effect is every important on the pattern formation of network. In this paper, a regular network of Hindmarsh–Rose neurons is designed in a two-dimensional square array with nearest-neighbor connection type. The neurons on the boundary are excited with random stimulus. It is found that spiral waves, even a pair of spiral waves could be developed in the network under appropriate coupling intensity. Otherwise, the spatial distribution of network shows irregular states. A statistical variable is defined to detect the collective behavior by using mean field theory. It is confirmed that regular pattern could be developed when the synchronization degree is low. The potential mechanism could be that random perturbation on the boundary could induce coherence resonance-like behavior thus spiral wave could be developed in the network.

Mode Veering and Internal Resonance in Mechanically Coupled Nanocantilevers under Electrostatic Actuation

We propose a multiphysics model to investigate the nonlinear dynamics of two coupled nanocantilevers under electrostatic actuation and 1:1 internal resonance (IR). We demonstrate that the region below the first pull-in instability is governed by the veering phenomenon between the modes. IR conditions are analyzed in the veering region and the modulation equations are derived and numerically solved. Remarkably, the first NEMS imposes its bifurcation topology to the second one and the energy transfer between the two oscillators can be controlled by the actuation voltages.

Bifurcation topology transfer in nonlinear nanocantilever arrays subject to parametric and internal resonances

The collective nonlinear dynamics of a coupled array of nanocantilevers is investigated while tak- ing into account the main sources of nonlinearities. The amplitude and phase equations of this device, subject to parametric and internal resonances, are analytically derived by means of a multi-modal Galerkin discretiza- tion coupled with a multiscale analysis. Based on the steady-state solutions of these equations, the frequency responses are numerically computed for a two-beam array. The effects of different parameters are investigated and several dynamical aspects are confirmed by numerical simulations. Particularly, we have demonstrated that the bifurcation topology transfer is imposed by the first nanocantilever and it can be general to the collective nonlinear dynamics of the NEMS array.

Forced large amplitude periodic vibrations of non-linear Mathieu resonators for microgyroscope applications

This paper describes a comprehensive non-linear multiphysics model based on the Euler–Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parametric analyses are presented in order to investigate the effect of the proof mass frequency on the bifurcation topology. The model shows that the optimal sensitivity is reached for resonant microgyroscopes designed with sensing frequency four times faster than the actuation one.

Stability control of nonlinear micromechanical resonators under simultaneous primary and superharmonic resonances

Fast effects of a slow excitation on the main resonance of a nonlinear micromechanical resonator are analytically and experimentally investigated. We show, in particular, how the bifurcation topology of an undesirable unstable behavior is modified when the resonator is simultaneously actuated at its primary and superharmonic resonances. A stabilization mechanism is proposed and demonstrated by increasing the superharmonic excitation.

Signal Amplification by Sensitive Control of Bifurcation Topology

We describe a novel amplification scheme based on inducing dynamical changes to the topology of a bifurcation diagram of a simple nonlinear dynamical system. We have implemented a first bifurcation-topology amplifier using a coupled pair of parametrically driven high-frequency nanoelectromechanical systems resonators, demonstrating robust small-signal amplification. The principles that underlie bifurcation-topology amplification are simple and generic, suggesting its applicability to a wide variety of physical, chemical, and biological systems.

Bifurcation topology tuning of a mixed behavior in nonlinear micromechanical resonators

We report the experimental observation of a four-bifurcation-point (or five possible amplitudes for a given frequency) behavior of electrostatically actuated micromechanical resonators, called the mixed (first hardening then softening) behavior. We also demonstrate both analytically and experimentally tuning the bifurcation topology of this behavior via an electrostatic mechanism. An analytical model allows for the qualitative as well as quantitative explanation of the experiments and serves as a simple tool for design of nonlinear micromechanical devices under high drive.