Bifurcation analysis of an electro-statically actuated micro-beam in the presence of centrifugal forces

In this paper, the influence of centrifugal forces on the stability of an electro-statically actuated clamped–clamped micro-beam has been investigated. The non-dimensional governing static and dynamic equations have been linearized using the step by step linearization method (SSLM), then, a Galerkin-based reduced order model has been used to solve the linearized equations. For constant value of a bias DC voltage and different values of angular velocity the equilibrium points of the corresponding autonomous system including stable center points, unstable saddle points and singular points have been obtained using the equivalent mass-spring model. Subsequently the bifurcation diagram has been depicted using the obtained fixed point. The static pull-in voltage value for different values of angular velocity and the static pull-in angular velocity for different values of bias voltage have been calculated. The obtained results are validated using results of previous studies and a good agreement has been observed. The effect of the centrifugal force on the fixed points has been studied using the phase portraits of the system for different initial conditions. Moreover, the effects of centrifugal forces on the dynamic pull-in behavior have been investigated using time histories and phase portraits for different angular velocities.     

Analysis of nonlinear dynamic behavior of electrically actuated micro-beam with piezoelectric layers and squeeze-film damping effect

An analytical model based on a nonlinear deflection equation and the Reynolds equation is proposed to describe the dynamic behavior of an electrically actuated micro-beam with two piezoelectric layers. The proposed model takes explicit account of the fringing field effect, the axial stress effect, the residual stress effect, and the squeeze-film damping effect between the micro-beam and the lower electrode. The nonlinear governing equation of the micro-beam is solved using a hybrid computational scheme comprising the differential transformation method and the finite difference method. The validity of the analytical model and numerical solution procedure is demonstrated by comparing the result obtained for the pull-in voltage of a micro-beam actuated by a DC voltage only with that presented in the literature. It is shown that the nonlinear dynamic response of the micro-beam can be controlled using a combined driving scheme consisting of both the magnitude and the frequency of the AC actuating voltage and a DC driving voltage. The effects of the AC/DC actuating conditions, micro-beam geometry parameters, and squeeze-film damping force on the center-point displacement of the micro-beam are systematically examined. In addition, the actuating conditions which ensure the stability of the micro-beam are identified by means of phase portraits and Poincaré maps. In general, the results show that the analytical model and hybrid numerical scheme provide a feasible means of analyzing the dynamic response of a variety of electrostatically-actuated microstructures.     

Application of piezoelectric actuation to regularize the chaotic response of an electrostatically actuated micro-beam

The impetus of this study is to investigate the nonlinear chaotic dynamics of a clamped–clamped micro-beam exposed to simultaneous electrostatic and piezoelectric actuation. The micro-beam is sandwiched with piezoelectric layers throughout its length. The combined DC and AC electrostatic actuation is imposed on the micro-beam through two upper and lower electrodes. The piezoelectric layers are actuated via a DC electric voltage applied in the direction of the height of the piezoelectric layers, which produces an axial force proportional to the applied DC voltage. The governing differential equation of the motion is derived using Hamiltonian principle and discretized to a nonlinear Duffing type ODE using Galerkin method. The governing ODE is numerically integrated to get the response of the system in terms of the governing parameters. The results show that the response of the system is greatly affected by the amounts of DC and AC electrostatic voltages applied to the upper and lower electrodes. The results show that the response of the system can be highly nonlinear and in some regions chaotic. Evaluating the K–S entropy of the system, based on several initial conditions given to the system, the chaotic response is distinguished from the periodic or quasiperiodic ones. The main objective is to passively control the chaotic response by applying an appropriate DC voltage to the piezoelectric layers.     

Analytical modeling of static behavior of electrostatically actuated nano/micromirrors considering van der Waals forces

In this paper,the effect of van der Waals(vdW)force on the pull-in behavior of electrostatically actuatednano/micromirrors is investigated.First,the minimum potential energy principle is utilized to find the equation governing the static behavior of nano/micromirror under electrostatic and vdW forces.Then,the stability of static equilibrium points is analyzed using the energy method.It is foundthat when there exist two equilibrium points,the smaller oneis stable and the larger one is unstable.The effects of different design parameters on the mirror’s pull-in angle andpull-in voltage are studied and it is found that vdW forcecan considerably reduce the stability limit of the mirror.Atthe end,the nonlinear equilibrium equation is solved numerically and analytically using homotopy perturbation method(HPM).It is observed that a sixth order perturbation approximation can precisely model the mirror’s behavior.The results of this paper can be used for stable operation design andsafe fabrication of torsional nano/micro actuators.     

Dynamic stability of electrostatic torsional actuators with van der Waals effect

The influence of van der Waals (vdW) force on the stability of electrostatic torsional nano-electro-mechanical systems (NEMS) actuators is analyzed in the paper. The dependence of the critical tilting angle and voltage is investigated on the sizes of structure with the consideration of vdW effects. The pull-in phenomenon without the electrostatic torque is studied, and a critical pull-in gap is derived. A dimensionless equation of motion is presented, and the qualitative analysis of it shows that the equilibrium points of the corresponding autonomous system include center points, stable focus points, and unstable saddle points. The Hopf bifurcation points and fork bifurcation points also exist in the system. The phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, as well as homoclinic orbits.     

The effect of time-delayed feedback controller on an electrically actuated resonator

This paper presents a study of the effect of a time-delayed feedback controller on the dynamics of a Microelectromechanical systems (MEMS) capacitor actuated as a resonator by DC and AC voltage loads. A linearization analysis is conducted to determine the stability chart of the linearized system equations as a function of the time delay period and the controller gain. Then the method of multiple-scales is applied to determine the response and stability of the system for small vibration amplitude and voltage loads. It is shown that negative time-delay feedback control gain can lead to unstable responses, even if AC voltage is relatively small compared to the DC voltage. On the other hand, positive time delay can considerably strengthen the system stability even in fractal domains. We also show how the controller can be used to control damping in MEMS, increasing or decreasing, by tuning the gain amplitude and delay period. Agreements among the results of a shooting technique, long-time integration, basin of attraction analysis with the perturbation method are achieved.     

Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions. The project supported by the National Natural Science Foundation of China (19972025)     

Effect of disturbances and sensorimotor deficits on the postural robustness of an ankle–hip model of balance on a balance board

This paper investigates the effect of postural disturbances and sensorimotor deficits on the robustness of the upright posture (UP) for a human body model balancing on a balance board (BB). The robustness is investigated by computing the gradient field along the basin of attraction (BoA) of an asymptotically stable equilibrium point. The human model is modeled as a double-inverted pendulum (hip and ankle joints). The human-BB system is assumed actuated by torques at the hip, BB hinge, and ankle joints. Postural disturbances induce an initial joint angle velocity either at the ankle or at the hip joint. Moreover, either proprioceptive or visual and vestibular deficits are considered in the human-BB model. The nonlinear dynamic equation of motion of the human-BB system is numerically solved to obtain the BB, ankle, and hip joint angle position and velocity profiles. The BoA of the human-BB UP equilibrium is built as the set of initial conditions whose resulting time-series profiles converge to the equilibrium. It was shown that UP is more robust to disturbances that induce hip joint initial angle velocity. That is probably due to the fact that disturbances that induce ankle joint initial velocity affect the whole body, while disturbances that induce hip joint angle initial velocity only affect the trunk. Whenever visual and vestibular deficits are considered, the UP is more robust if proprioceptive gain and BB stiffness are small. Contrarily, whenever proprioceptive deficits are considered, the UP is more robust if visual and vestibular gain and BB stiffness are large. The method proposed here (the BoA and the gradients) can be used to systemically provide understanding about the robustness of the human-BB UP to external disturbances, which may help to identify people with a higher risk of fall.     

Dynamic pull-in phenomenon in MEMS resonators

We study the pull-in instability in microelectromechanical (MEMS) resonators and find that characteristics of the pull-in phenomenon in the presence of AC loads differ from those under purely DC loads. We analyze this phenomenon, dubbed dynamic pull-in, and formulate safety criteria for the design of MEMS resonant sensors and filters excited near one of their natural frequencies. We also utilize this phenomenon to design a low-voltage MEMS RF switch actuated with a combined DC and AC loading. The new switch uses a voltage much lower than the traditionally used DC voltage. Either the frequency or the amplitude of the AC loading can be adjusted to reduce the driving voltage and switching time. The new actuation method has the potential of solving the problem of high driving voltages of RF MEMS switches.     

Stability and oscillations in a slow-fast flexible joint system with transformation delay

Flexible joints are usually used to transfer velocities in robot systems and may lead to delays in motion transformation due to joint flexibility. In this paper, a linkrotor structure connected by a flexible joint or shaft is firstly modeled to be a slow-fast delayed system when moment of inertia of the lightweight link is far less than that of the heavy rotor. To analyze the stability and oscillations of the slowfast system, the geometric singular perturbation method is extended, with both slow and fast manifolds expressed analytically. The stability of the slow manifold is investigated and critical boundaries are obtained to divide the stable and the unstable regions. To study effects of the transformation delay on the stability and oscillations of the link, two quantitatively different driving forces derived from the negative feedback of the link are considered. The results show that one of these two typical driving forces may drive the link to exhibit a stable state and the other kind of driving force may induce a relaxation oscillation for a very small delay. However, the link loses stability and undergoes regular periodic and bursting oscillation when the transformation delay is large. Basically, a very small delay does not affect the stability of the slow manifold but a large delay affects substantially.     

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.     

Effects of time delayed velocity feedbacks on self-sustained oscillator with excitation

Both the primary resonant solutions and their bifurcations due to time delayed velocity feedbacks used in a self-sustained oscillator with excitation were further investigated. A model was proposed by adding linear and nonlinear time delayed feedbacks to a representative non- autonomous system ( with external forcing ). The stability condition of the linearized system at trivial equilibrium was discussed, which leads to a critical stability boundary where periodic solutions may occur. The main attention was focused on bifurcations from the primary resonant solutions. It is found that the stable primary resonant solution may appear periodically in the time delay. Meanwhile, the unstable regions for such solutions are also obtained, predicting the occurrence of quasi-periodic motions.     

An experimental study of bifurcation,chaos, and dimensionality in a system forced through a bifurcation parameter

An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.     

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.     

参-强激励联合作用下输流管的分岔和混沌行为研究

研究输送脉动流的两端固定输流管道在其基础简谐运动激励下的分岔和混沌行为,考虑管道变形的几何非线性和管道材料的非线性因素,推导了系统的非线性运动方程,并应用Galerkin方法对其进行了离散化处理。通过采用数值模拟方法,对系统的运动响应进行仿真,重点探讨了流体平均流速、流速脉动振幅以及基础简谐运动激励振幅对系统动态特性的影响。结果表明,系统在不同的参数下会发生围绕不同平衡点的周期和混沌等运动,并在系统中发现了两条通向混沌运动的途径:倍周期分岔和阵发混沌运动。     

Three-dimensional dynamics of supported pipes conveying fluid

This paper deals with the three-dimensional dynamics and postbuckling behavior of flexible supported pipes conveying fluid, considering flow velocities lower and higher than the critical value at which the buckling instability occurs. In the case of low flow velocity, the pipe is stable with a straight equilibrium position and the dynamics of the system can be examined using linear theory. When the flow velocity is beyond the critical value, any motions of the pipe could be around the postbuckling configuration (non-zero equilibrium position) rather than the straight equilibrium position, so nonlinear theory is required. The nonlinear equations of perturbed motions around the postbuckling configuration are derived and solved with the aid of Galerkin discretization. It is found, for a given flow velocity, that the first-mode frequency for in-plane motions is quite different from that for out-of-plane motions. However, the second- or third-mode frequencies for in-plane motions are approximately equal to their counterparts for out-of-plane motions, keeping almost constant values with increasing flow velocity. Moreover, the orientation angle of the postbuckling configuration plane for a buckled pipe can be significantly affected by initial conditions, displaying new features which have not been observed in the same pipe system factitiously supposed to deform in a single plane.     

Suppression of chaos in a MEMS resonator by delayed position feedback

A delayed position feedback control is applied on DC voltage source for suppressing chaos of a typical MEMS resonator actuated by electrostatic forces. A theoretical necessary condition for chaotic oscillation of the controlled system is presented. Numerical results and the analytical prediction reveal the evolution of dynamical behavior of the system with AC voltage amplitude and the control effect of delayed feedback on reducing chaos of the system. It shows that the delayed feedback control is effective on suppressing chaos of the micro mechanical resonator.     

Dynamic instability of vibrating carbon nanotubes near small layers of graphite sheets based on nonlocal continuum elasticity

This article presents a new asymptotic method to predict dynamic pull-in instability of nonlocal clamped–clamped carbon nanotubes (CNTs) near graphite sheets. Nonlinear governing equations of carbon nanotubes actuated by an electric field are derived. With due allowance for the van der Waals effects, the pull-in instability and the natural frequency–amplitude relationship are investigated by a powerful analytical method, namely, the parameter expansion method. It is demonstrated that retaining two terms in series expansions is sufficient to produce an acceptable solution. The obtained results from numerical methods verify the strength of the analytical procedure. The qualitative analysis of system dynamics shows that the equilibrium points of the autonomous system include center points and unstable saddle points. The phase portraits of the carbon nanotube actuator exhibit periodic and homoclinic orbits.     

非自治时滞反馈控制系统的周期解分岔和混沌

研究时滞反馈控制对具有周期外激励非线性系统复杂性的影响机理,研究对应的线性平衡态失稳的临界边界,将时滞非线性控制方程化为泛函微分方程,给出由Hopf分岔产生的周期解的解析形式．通过分析周期解的稳定性得到周期解的失稳区域,使用数值分析观察到时滞在该区域可以导致系统出现倍周期运动、锁相运动、概周期运动和混沌运动以及两条通向混沌的道路：倍周期分岔和环面破裂．其结果表明,时滞在控制系统中可以作为控制和产生系统的复杂运动的控制“开关”．     

Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed,variable axial tension under two-frequency parametric excitations and internal resonance

This paper investigates nonlinear combined parametric transverse vibrations of a traveling viscoelastic beam. The combined parametric excitations originate from the time dependency of axial velocity as well as axial tension. Two parametric excitations are enforced into the system amid the internal resonance. Two-frequency parametric resonance is assumed to be comprised of combination parametric resonance of first two modes due to the time dependency of axial velocity, and the principal parametric resonance of first mode due to the variable tension in the axial direction in the presence of internal resonance for viscoelastic beam is considered for the first time. The higher-order integro-partial differential equation of motion is solved through direct method of multiple scales. Continuation algorithm is employed to explore the stability and various bifurcations of the nonlinear dynamic system. Focus has been made to study the effect of variations of fluctuating tension component, fluctuating velocity component independently and when combined, internal and parametric frequency detuning parameters and damping on the system response. Frequency response equilibrium curves are complex and unique in shapes which are embodied with various bifurcations. Such steady-state behavior is not seen in the existent literature. With variation in fluctuating velocity component, the number of steady-state nontrivial equilibrium curves increases to three and with variation in fluctuating axial tension, they become four. In this process, significant changes in stability, number and position of various bifurcations like supercritical and subcritical pitchfork, Hopf and saddle node are observed. Unlike the previous study, the shape, stability and bifurcations of equilibrium curves under the combined effect of axial velocity and tension closely match with the case of fluctuating axial tension component. The effect of variation in internal and parametric frequency detuning parameter is more realized for second mode compared to first mode. A comparison of the present work with a previous one where axial tension is variable reveals many qualitative and quantitative similarities and dissimilarities. But when compared with earlier work where axial velocity is constant, significant dissimilarities are surfaced. The system displays a wide ranging dynamic behavior including stable periodic, quasiperiodic and unstable chaotic behavior. The numerical computation depicts various nonlinear characteristics and oscillatory behaviors which are not found so far in the existent literature.