Parametric Excitation Frequency Research Articles

Theoretical modeling and experimental investigation of in-phase resonant MEMS mirrors with cascaded structures

This paper proposes an efficient nonlinear one-dimensional (1D) compact mass-damper-spring model to predict the dynamic response of electrostatic resonant micro-electro-mechanical system (MEMS) mirrors with cascaded structures. The time-dependent damping moment due to viscous shear and pressure drag is computed using semi-empirical analytical equations for comb-drive structures and device frames. Nonlinear electrostatic force induced by the comb drives is efficiently acquired based on the hybrid method. The optimized device is fabricated using MEMS fabrication processes based on a 4-inch silicon-on-insulator wafer. The proposed compact model with the measured key parameters from the fabricated device shows excellent capability to accurately predict nonlinear dynamic responses of the fabricated device, including parametric excitation and hysteretic frequency response, with an average error of less than 5%. In particular, our 1D model is three orders of magnitude faster than the conventional finite element method model (0.8 s versus 1 h), enabling efficient system-level optimization of the critical design parameters. Based on the parametric study, electrode gap distance and torsion spring width are found to be two critical design parameters and dimensional analysis is conducted for design optimization with scan angle enhancing from 16.8° to 24° compared with the first design.

Strange attractors in a parametrically excited nonlinear oscillator

Nonlinear oscillators exhibit rich and complex dynamic behaviour and are highly sensitive to initial conditions. The current paper studies the existence of strange attractors in a nonlinear oscillator namely the Duffing oscillator under harmonically varying parametric excitation. Numerical simulation is carried out via fourth-order Runge-kutta algorithm to study the presence of strange attractors in a parametrically excited nonlinear oscillator for different values of control parameters viz. coefficient of damping, frequency and amplitude of parametric excitation. For amplitude of parametric excitation d = 6.25 and frequency of parametric excitation Ω = 0.5, with coefficient of damping c = 0.2, the phase plane trajectory and time history show existence of classical chaotic attractors. There is oscillation between two chaotic attractors in a double well potential for every period of excitation. For amplitude of parametric excitation d = 6.25 and frequency of parametric excitation Ω = 0.07, with coefficient of damping c = 0.2, the system also shows the presence of two co-existing fixed point chaotic attractors which is found to merge with change in the parameter values. The dynamical behaviour of the fixed point chaotic attractors are found to be different from that of classical chaotic attractors.

Parametric excitation analysis for system performance of piezoelectric energy harvesters

Capitalizing on the advantages of parametric excitation and real-life low frequency vibration, the steady-state response of piezoelectric energy harvesting system composing of a cantilever beam with a tip mass and horizontal excitation at the fixed end is investigated. By applying Hamilton's principle, the nonlinear partial differential equation of a cantilever bimorph piezoelectric energy harvesting system with an additional tip mass is derived and analyzed. The cantilever is modeled as an axially non-elongated Euler Bernoulli beam with geometric and damping nonlinearities. By using the Galerkin method, the nonlinear partial differential equation is reduced to an electromechanical coupling system that governs a cantilever piezoelectric energy harvesting system with a tip mass under parametric excitation. The first-order resonance response of this harvesting system is studied by using the method of multiple scales. The analytical response expressions and first-order amplitude functions for vertical displacement, output voltage and output power are derived and analyzed. The influence of different impedance and tip mass on the piezoelectric energy harvesting system performance is summarized and concluded. For the parametric excitation system, the load resistance shows significant influence on the initial threshold of the piezoelectric energy harvesters under parametric excitation. For a short circuit, the initial threshold increases with increasing load resistance. For an open circuit, the initial threshold decreases with increasing load resistance. For an open circuit resistance or a short circuit resistance under parametric excitation, an increase in the end block mass effectively reduces the resonance frequency of the piezoelectric energy harvesters. Consequently, the energy harvesting bandwidth is increased, and the harvesting effect of the piezoelectric energy harvesters is enhanced. In this paper, a more accurate and practical theoretical model is established to predict the electromechanical coupling behavior of cantilever piezoelectric energy harvester under parametric excitation. Using this approach, it is possible to complement parametric and direct excitation such that the advantages of parametric excitation can be achieved. It is also significant to improve the energy conversion efficiency of energy harvesting system.

Multi-Frequency Homotopy Analysis Method for Coupled Van der Pol-Duffing System with Time Delay

This paper mainly studied the analytical solutions of three types of Van der Pol-Duffing equations. For a system with parametric excitation frequency, we knew that the ordinary homotopy analysis method would be unable to find the analytical solution. Thus, we primarily used the multi-frequency homotopy analysis method (MFHAM). First, the MFHAM was introduced, and the solution of the system was expressed by constructing auxiliary linear operators. Then, the method was applied to three specific systems. We compared the numerical solution obtained using the Runge–Kutta method with the analytical solution to verify the correctness of the latter. Periodic solutions, with and without time delay, were also compared under the same parameters. The results demonstrated that it was both effective and correct to use the MFHAM to find analytical solutions to Van der Pol-Duffing systems, which were classical systems. By comparison, the MFHAM proved to be effective for time delay systems.

Nonlinear interaction of parametric excitation and self-excited vibration in a 4 DoF discontinuous system

Interaction between parametric excitation and self-excited vibration has been subjected to numerous investigations in continuous systems. The ability of parametric excitation to quench self-excited vibrations in such systems has also been well documented. But such effects in discontinuous systems do not seem to have received comparable attention. In this article, we investigate the interaction between parametric excitation and self-excited vibration in a four degree of freedom discontinuous mechanical system. Unlike majority of studies in which oscillatory nature of stiffness accounts for parametric excitation, we consider a much more practical case in which parametric excitation is provided by a massless rotor of rectangular cross section with a cylinder-like mass concentrated at the center. The rotor arrangement is placed on a friction-induced self-excited support in the form of a frame placed on a belt moving with constant velocity. This frame is connected to a supplementary mass. A Stribeck friction model is considered for the mass in contact with the belt. The frictional force between the mass and the belt is oscillatory in nature because of the variation of normal force due to parametric excitation from the rotor. Our investigations reveal mutual synchronization of parametric excitation and self-excited vibration in the system for specific parameter values. The existence of a stable limit cycle with constant synchronized fundamental frequency, for a range of parametric excitation frequencies, is established numerically. Investigation based on frequency spectra and Lissajous curves reveals complex synchronization patterns owing to the presence of higher harmonics. The system is also shown to exhibit Neimark–Sacker bifurcations under the variation of belt velocity. Furthermore, variation in belt velocity and coupling stiffness is seen to cause a breakup of quasi-periodic torus with small-amplitude oscillations to form large amplitude chaotic orbits. This points toward the possibility of vibration suppression in the system by tuning the parameters for stabilizing the small-amplitude quasi-periodic response. An example of co-existence of different attractors in the system is also presented.

Bifurcation dynamics of 1DOF parametric oscillator with stiffness-hardening characteristic and dry friction

In this work, we have mathematically modeled a 1DOF parametric oscillator with stiffness-hardening characteristics and dry friction and investigated both experimentally and numerically the bifurcation dynamics. The system consists of a cart moving along a linear rolling guide widely used in the industry. It has a stiffness comprised of two components: a linear time-variable part generated by a rotating rod of a rectangular cross-section and a nonlinear hardening stiffness caused by magnetic springs. In the case of nonlinear resistance of motion in a rolling bearing, regardless of the true nature of this phenomenon, it was modeled as the sum of viscous damping and the second component mathematically equivalent to dry friction. The trivial solution was observed to be stable in the whole range of parametric excitation frequencies. But there is a frequency range where the system is bistable, a periodic attractor coexists with the stable equilibrium position, and the branch of the periodic orbit is isolated, i.e., not connected with the equilibrium position, and forms an Isola. This distinguishes the analyzed system from the commonly investigated parametric oscillators, including the classical Mathieu equation and its various versions. Our work perfectly agreed between the numerical simulations and the experimental data. Moreover, the mathematical model for different friction values is investigated, showing the transition between the system without dry friction and the actual rig. The stability of the equilibrium position is tested, and the bifurcation dynamics of periodic orbits are presented using numerical continuation methods, obtaining complete agreement between the results obtained using different ways.

Forced Response Approach on Parametric Vibration with Time Periodic Coefficient

In this study, based on Galerkin discretization and frequency splitting, the forced vibration response of a large span cable with time periodic tension is approached as a closed-form solution, i.e. a special trigonometric series with mode functions. According to the principle of modulation feedback, the forced vibration response solution can be expressed as a series of linear combination of external excitation frequency and parametric excitation frequency with modes. Using orthogonality relation, the formula of forced vibration response approach is obtained. Investigations show that the presented approach computes the forced vibration response with higher accuracy. Therefore, this study is available to analyze the vibration of a large span cable with time periodic tension, and the characteristics of vibration response could be used as an important information source for the failure diagnosis and the health monitoring of cable structure.

An incremental harmonic balance method with two time-scales for quasi-periodic responses of a Van der Pol–Mathieu equation

An incremental harmonic balance (IHB) method with two time-scales is presented and used for calculating accurate quasi-periodic responses of a one-degree-of-freedom Van der Pol–Mathieu equation with coupled self-excited vibration and parametrically excited vibration with 1:2 resonance. For periodic responses of the Van der Pol–Mathieu equation with only one basic frequency, the traditional IHB method is used to automatically trace their nonlinear frequency response curves. Stability and bifurcations of the periodic responses for given parameters are then determined by the Floquet theory using the precise Hsu’s method. It is found that a jump from a periodic response to a quasi-periodic response at a critical point results from a saddle node bifurcation. For quasi-periodic responses of the Van der Pol–Mathieu equation, their spectra contain uniformly spaced sideband frequencies that have not been observed heretofore, which involve two incommensurate basic frequencies, i.e., the parametric excitation frequency and a priori unknown frequency related to uniformly spaced sideband frequencies. The IHB method with two time-scales is formulated to deal with cases where one basic frequency is unknown a priori, in order to automatically trace nonlinear frequency response curves of quasi-periodic responses of the Van der Pol–Mathieu equation with 1:2 resonance and accurately calculate all frequency components and their corresponding amplitudes even at critical points. Results of the Van der Pol–Mathieu equation obtained from the IHB method with two time-scales are in excellent agreement with those from numerical integration using the fourth-order Runge–Kutta method. This investigation reveals rich dynamic characteristics of the Van der Pol–Mathieu equation in a wide range of parametric excitation frequencies.

High-frequency vibrational control of principal parametric resonance of a nonlinear cantilever beam: Theory and experiment

The effects of high-frequency parametric excitation on the principal parametric resonance of a nonlinear beam are studied both theoretically and experimentally. Only the first mode of vibration under simultaneous slow-fast parametric excitation is considered for investigation. The method of direct partition of motion yields slow dynamics of the system which is further analyzed by multiple time scale method to obtain the amplitude and phase response of the system. It is observed that the effective damping and stiffness of the system is significantly modified by the high frequency parametric excitation. The analysis reveals that the parametric resonance curve shifts towards the higher range of slow parametric excitation frequency due to high-frequency excitation. The peak resonant response decreases with the increasing value of the strength of the high frequency excitation and also beyond a critical value of the strength of the high-frequency excitation, the jump resonance is eliminated. Overall, it is found that the principal Parametric resonance can be effectively controlled and suppressed by high frequency excitation. Analytical results are validated by direct numerical simulations and experiments. For some parameter values, vibrational resonance is also observed in theory and experiment. The parametric vibrational resonance of a cantilever beam is not previously observed experimentally and thus, it is a new finding of the present study. The vibrational resonance is believed to be utilized in amplifying the response of parametrically driven microcantilever beams found in many microsystems applications.

Resonance analysis of a single-walled carbon nanotube

Abstract To improve the reliability of carbon nanotube sensors is one of the key problems when they are applied to engineering practice. Thus, it is of great theoretical significance and practical engineering application value to study the influence of structural parameters and working environment of micro-nanosensor on the dynamic stability and complex behavior of nonlinear system involved in nanotube devices. In this paper, resonance behaviors of a single wall carbon nanotubes (SWCNT) with parametric excitation and external excitation is investigated under quasi-periodic perturbation. The global dynamics of the unperturbed system is explored, then the 1:2 subharmonic resonance and 2:1 superharmonic resonance are analyzed by using the second averaging method with the assumption that the parametric excitation frequency and external excitation frequency are irrational. Subsequently, the global dynamics of the unperturbed averaged systems are investigated by using Poincare compactification theories. The criteria for the existence of homoclinic Smale chaos of the approximate system with periodic perturbation is obtained by using the homoclinic Melnikov method. Finally, the numerical simulations including the bifurcation diagrams, Lyapunov exponent spectrums, various orbits (period-1, period-2, chaotic) and Poincare sections are presented to verify the theoretic results.

Dynamics of an articulated chain of rigid pipes discharging fluid under concomitant support excitation: A numerical analysis

This work proposes a numerical investigation on the occurrence of parametric excitation in the classical Benjamin problem, when the support of the articulated chain of pipes is subjected to a prescribed oscillatory motion. Equations of motion for an articulated chain composed of two rigid pipes ejecting fluid are derived. Floquet theory is applied to the linearized mathematical model, and curves of instability as functions of the parametric excitation (amplitude and frequency) are obtained for selected internal flow velocities. In addition to the linear analysis, maps (showing the post-critical response as a function of some control parameters) are computed using numerical integrations of the nonlinear model. Among other findings, this paper reveals that the presence of the internal flow significantly affects the stability of the trivial solution. Furthermore, the concomitant parametric excitation and internal flow effects lead to maps of post-critical response with a marked erosion, i.e., small variations in the parameters of the mathematical model have significant effects in the observed response.

Parametric amplification performance analysis of a vibrating beam micro-gyroscope with size-dependent and fringing field effects

In this study, a parametric amplification model of a vibrating beam micro-gyroscope is established with the effects of size-dependent and fringing field taken into account. Compared with conventional electrostatic beam gyroscopes, piezoelectric films are added to the four surfaces of the rectangular beam of vibrating micro-gyroscopes. Due to piezoelectric effect, the beam will be subjected to an axial excitation, which provides parametric amplification to enlarge the transverse vibration and to enhance the sensitivity. The effects of the size-dependent, fringing field and piezoelectric voltages on static deflection, eigenvalue analysis and dynamic amplification of the vibrating beam micro-gyroscopes are investigated. The influence of damping on the stability analysis of the gyroscopic system is also studied. When the frequency of parametric excitation is twice the natural frequency of the system, it is found that higher piezoelectric AC voltage amplitudes within the stable region are given in both sense and drive directions, which may generate a better sensitivity of the vibrating beam micro-gyroscope.

Response of linear parametric amplifiers with arbitrary direct and parametric excitations

The effect of parametric amplification, where parametric excitation is added to direct excitation, on the performance of a dynamical system is investigated. To model the system, the linear forced Mathieu equation is considered. In order to obtain the response of the system without restrictions such as requiring the system parameters to be small or the parametric excitation frequency to be exactly twice the direct excitation frequency, the method of varying amplitudes (MVA) is used. Employing the MVA, the response of the system and its maximum value for arbitrary values of the system parameters including the excitation frequencies are derived. It is shown that when the ratio between the excitation frequencies is rational, the maximum response of the system is highly dependent on the phase angle between the excitations. However, when the frequency ratio is irrational, the maximum response of the system is not dependent on the phase angle. Consequently, large values of response can be attained by using an irrational frequency ratio regardless of the phase angle. MVA results for peak responses show good agreement with numerical results obtained by direct integration of the equation of motion.

Computational technique to free vibration response in a multi-degree of freedom parametric system

The free vibration response is helpful to understand the dynamic characteristics of parametric system, and further could be used as important information source for the failure diagnosis and health monitoring of structure. In this paper, the free vibration response of a multi-degree of freedom (MDOF) parametric system is approached as a closed-form solution, a special trigonometric series with a linear combination of the principal oscillation frequency and the parametric excitation frequency. By algebra operation of harmonic balance, the parametric equation can be transformed into a linear homogeneous algebraic equation, on which the characteristic equation is determined and some important solutions are given. Main contribution of the presented technique is that it not only solves the computational problem of principal oscillation frequencies and modes with high computational accuracy, but also gives the splitting mode, which is the amplitude ratio of the combined harmonic resonance in the parametric system. Besides, the mathematical approximation reflects the physic characteristics in the free vibration response of parametric system. An example of real life is used to illustrate the applicability of the presented computational technique. The more expansion terms in approximation series, the smaller the computational error. Therefore, the given approximation provides a powerful computational tool for the free vibration response in the MDOF parametric system.

On the response attainable in nonlinear parametrically excited systems

The response of nonlinear parametrically excited systems is investigated. It is shown that perturbation methods including the method of multiple scales and the averaging method using the Krylov–Bogoliubov technique can provide precise predictions of the behavior of the system in the neighborhood of the principal parametric resonance. However, increasing the excitation frequency, they imply an infinite growth of the vibration amplitudes, which contradicts numerical and practical findings. To tackle this problem, the method of varying amplitudes (MVA) is used. Employing the MVA, an expression for the upper bound to the displacement response of the system is derived. The parametric excitation frequency, at which this response is attained, is defined explicitly. MVA results considering the first harmonic are identical to the results obtained by the first approximation of the Mitropolskii technique and show good agreement with numerical results obtained from direct integration of the equation of motion. Taking the first and third harmonics into account, MVA shows an excellent capability to capture the frequency at which the upper bound response is attained: the difference between the upper bound to the displacement response obtained from the MVA and the one obtained from numerical integration is less than 0.1%.

Parametrically Excited Nonlinear Two-Degree-of-Freedom Electromechanical Systems

This paper presents a nonlinear parametrically excited cantilever beam with electromagnets. A parametrically excited two-degree-of-freedom (2-DOF) system with linear time-varying stiffness, nonlinear cubic stiffness, nonlinear cubic parametric stiffness and nonlinear damping is considered. In previous studies the stability and bifurcation of the nonlinear parametrically excited 2-DOF were investigated through analytical, semi-analytical and numerical methods. Unlike previous studies, in this contribution the system’s response amplitude and phase at parametric resonance and parametric combination resonance are demonstrated experimentally and some novel results are discussed. Experimental and analytical amplitude-frequency plots are presented to show the stable solutions. Solutions for the system response are presented for specific values of parametric excitation frequency and the energy transfer between modes of vibrations is observed. The results presented in this paper prove that the bifurcation point and hence the bandwidth of the parametric resonance can be predicted correctly with the proposed analytical method. The proposed nonlinear parametrically excited 2-DOF can be used to design Micro ElectroMechanical Systems (MEMS) actuators and sensors. Validating the experimental results with the theory can improve the efficiency of these electrical systems.

Nonlinear forced vibration of functionally graded Timoshenko microbeams with thermal effect and parametric excitation

In this paper, the nonlinear forced vibration of functionally graded (FG) Timoshenko microbeams under thermal effects and parametric excitation is studied using von Kármán nonlinear theory, Hamilton's principle and the modified couple stress theory. The coupled nonlinear differential equations of transverse vibration modes are obtained by the static condensation and Galerkin scheme. The nonlinear algebraic equations of steady-state responses are presented based on the method of multiple scales, and solved using Newton iterative method. Finally, the internal resonance and principal resonance are proposed, and the influence of system parameters, involving the material parameter, length scale parameter, frequency of parametric excitation and thermal loading on the frequency-response is investigated. The reliability of presented Timoshenko microbeam model is confirmed by comparison with previously published results.

Control of nonlinear systems exhibiting chaos to desired periodic or quasi-periodic motions

A novel method in the design of controllers to drive general nonlinear systems to desired periodic or quasi-periodic motions is presented in this paper. The viability of the approach is demonstrated by controlling chaotic systems to desired motions. The proposed control system consists of a combination of a nonlinear feedforward controller and a linear feedback controller. The control gains for the feedback controller are determined by performing the stability analysis of the closed-loop systems that contain periodic or quasi-periodic coefficients. For the case of periodic coefficients, stability is determined using the well-known Floquet theory. Since there is no rigorous mathematical theory for the analysis of quasi-periodic systems, an approximate technique has been applied to determine the stability where the coefficients turned out to be quasi-periodic. In this approach, a quasi-periodic system is replaced by an approximate periodic system with an appropriate large principal period such that Floquet theory can be applied. State transition matrices for these closed-loop systems are computed symbolically in terms of the unknown control gains using shifted Chebyshev polynomials and Picard iterations. Stability diagrams obtained from the symbolic approach are used to select control gains to guarantee asymptotic stability of the feedback systems. Three examples of chaotic systems are studied in order to show applicability to a diverse class of problems. In the first case, the chaotic motion of a forced Duffing oscillator is driven to a two-frequency quasi-periodic square wave and a fixed point. In the control to a fixed point, the system is constrained to follow a logarithmic spiral trajectory. The second case involves the control of the chaotic attractor of a parametrically forced Lorenz system to a periodic orbit whose frequency is irrationally related to the parametric excitation frequency. In the last case, the chaotic behavior of a Mathieu–Duffing oscillator is successfully driven to a two-frequency quasi-periodic motion.

Studies on the dynamic stability of an axially moving nanobeam based on the nonlocal strain gradient theory

In this paper, we studied the parametric resonance issue of an axially moving viscoelastic nanobeam with varying velocity. Based on the nonlocal strain gradient theory, we established the transversal vibration equation of the axially moving nanobeam and the corresponding boundary condition. By applying the average method, we obtained a set of self-governing ordinary differential equations when the excitation frequency of the moving parameters is twice the intrinsic frequency or near the sum of certain second-order intrinsic frequencies. On the plane of parametric excitation frequency and excitation amplitude, we can obtain the instability region generated by the resonance, and through numerical simulation, we analyze the influence of the scale effect and system parameters on the instability region. The results indicate that the viscoelastic damping decreases the resonance instability region, and the average velocity and stiffness make the instability region move to the left- and right-hand sides. Meanwhile, the scale effect of the system is obvious. The nonlocal parameter exhibits not only the stiffness softening effect but also the damping weakening effect, while the material characteristic length parameter exhibits the stiffness hardening effect and damping reinforcement effect.

Rotor-pendulum self-synchronization of parametrical vibration generators on an isotropic elastic foundation

Significant improvement of dynamic characteristics of vibration machines on the basis of the use of resonant oscillatory systems with two or more degrees of freedom is proposed. It is shown that reasonable complication of models due to the increase in the number of degrees of freedom, taking account of nonlinearity makes it possible to increase reciprocal enhancement of vibrations of partial subsystems and to open additional opportunities in the development of new equipment and technologies. The involvement of objects of an oscillatory system in collective interaction is achieved on the basis of using compound resonances which occur in coupled systems only. Combination parametric resonance caused by pair-wise interaction of free modes of oscillations represents this kind of resonance. The results of theoretical and experimental studies of self-synchronization of parametric rotor-pendulum vibration generators installed on a common elastic isotropic foundation are presented in the paper. The dynamic model of a vibration machine is represented by a set of peer interacting nonlinear oscillators (pendulums) under resonant excitement of which the Huygens's effect, that is, pendulum clock-type self-synchronization (pendulum self-synchronization) occurs. One or several oscillators perform the functions of the working body of the vibration machine. Other oscillators of this system act as the inertial element of a rotor-pendulum vibration generator. It is shown that in the case of combined action of at least two rotors-pendulum vibration generators unbalanced rotor-type self-synchronization (rotor self-synchronization) takes place. Thus, the device under examination at the same time combines rotor and pendulum self-synchronization. The results of numerical modeling in the form of amplitude-frequency characteristics and dependences of generation frequencies on the frequency of parametric excitation are presented. It is established that the amplitude of the working body oscillations, due to resonant interactions with the generator pendulums, underreacts to an increase in damping. The sum of natural frequencies of pendulums and the working body meets the condition of combination parametric resonance over the whole range of the instability region.