Superharmonic Resonance Research Articles

Resonance capture and targeted energy transfer for suppressing aeroelastic instability of 2-D wing

Numerical simulations were conducted to study flow-induced vibration of a two-dimensional airfoil with two nonlinear energy sinks (NES). The relationship between targeted energy transfer (TET) and vibration suppression is analyzed in detail. The main system has two degrees of freedom – the pitch and heave. The two NES are treated as subsystems, in which the first NES is place at the leading edge and the second NES is placed at the trailing edge. The limit cycle oscillation (LCO), which is to be suppressed by the NES, is studied from the viewpoint of the TET. The resonance capture (RC) in the coupled nonlinear system is also discussed by the means of the energy and spectrum analysis. This is followed by a detailed target energy transfer discussion of the heave and pitch modes and the NES. In addition, the empirical mode decomposition (EMD) is utilized to obtain an intrinsic mode function (IMF) to analyze resonance capture in the system. The results show that the NES can absorb vigorous amount of energy from one of the specified vibration modes. As the RC occurs, the TET between the vibration modes in the coupled system becomes more significant. In particular, the TET between the NES and the wing becomes more efficient. This results in an increase in the critical freestream velocity as the NES suppresses the nonlinear vibration of the main system in a very effective way. As the total energy exceeds the suppression range of the subsystem, the NES loses its effectiveness on vibration suppression effect on the main system. The IMF of the EMD exhibits special super-harmonic resonance and frequency competition characteristics.

Nonlinear modes of piecewise linear systems forced vibrations close to superharmonic resonances

Method for strongly nonlinear piecewise linear systems forced vibrations analysis close to superharmonic resonances is suggested. The combination of Shaw–Pierre nonlinear modes and the Rauscher's approach is the foundation of this method. The superharmonic torsional vibrations of the power transmission are analyzed by using the suggested approach. The properties of the resonance superharmonic vibrations are treated.

Experimental and analytical study on the superharmonic resonance of size-dependent cantilever microbeams

The superharmonic resonance (SR) of size-dependent cantilever microbeams is investigated experimentally and analytically. Nickel cantilever microbeams are employed with concentrated harmonic force on the tip. The SR of order two, three, four, five, and six are observed. The frequency–response curves (FRCs) near the SR frequencies as well as the time histories are obtained. The FRCs indicate that the superharmonic resonant frequencies are different from the classical situation. Furthermore, a nonlinear model within the framework of modified couple stress theory is derived to interpret the observations by aid of Hamilton's variation principle. The resulting partial differential equation of motion is discretized into a series of ordinary differential equations (ODEs) by a two-mode Galerkin scheme. The ODEs are then solved analytically with the multi-dimensional Lindstedt–Poincaré method. Analytical results are in good agreement with experimental results in SR of order three. The effects of different nonlinear terms, length scale parameter, and damping coefficients on the nonlinear system are then discussed.

Dynamic and steady analysis of a 2-DOF vehicle system by modified incremental harmonic balance method

In this study, the dynamic response and stability of two-degree-of-freedom vehicle system subjected to an external excitation force with quadratic and cubic nonlinearities simultaneous are researched, and the differential equations of motion are deduced by utilizing the mechanical constitutive relationship. The accuracy of the model and the efficiency of modified incremental harmonic balance method are compared and verified with literature and numerical simulation results. Then the parameter researches of the system are carried out to investigate the influences of the mass ratio, excitation amplitude and nonlinear stiffness ratio of tire on the nonlinear characteristics and steady-state responses by amplitude–frequency curves, which have different level of influences on the dynamic responses of the vehicle system. Depending on the different control parameters, the system displays rich and varied behaviors including the jump discontinuity, multiple solution properties, super-harmonic resonance and softening/hardening-type nonlinearity, which are attributed to the strong nonlinearity and dynamical coupling characteristics of the vehicle system. Besides, the two remarkable resonance peaks and different branching paths between steady-state solution and unsteady-state solution are observed. The contributions of this study can provide valuable information to optimize and control the vehicle system, and be helpful for knowing more about nonlinear dynamic behaviors and stability of the system.

Research on geometric features of phase diagram and crack identification of cantilever beam with breathing crack

Geometric features of the phase diagram of a cantilever beam with a breathing crack are investigated. Influences of crack properties on the phase diagram geometry are studied and a method for crack identification based on phase diagram features is proposed. The pixel method is used to estimate different phase diagram areas. Multiple crack identification indexes are defined under the conditions of primary and super-harmonic resonance. The effects of crack location and depth on each index are analyzed. The contour lines for multiple identification indexes are used to estimate crack parameters. Numerical calculations indicate that the phase diagrams for super-harmonic resonance are more sensitive to crack parameters and that the proposed method is both feasible and effective.

Nonlinear dynamic analysis of dielectric elastomer minimum energy structures

This paper presents a theoretical study of nonlinear dynamic behavior of a dielectric elastomer minimum energy structure (DEMES). While planar Dielectric Elastomer Actuators (DEA) solely poses material nonlinearity in their model, geometrical nonlinearity imposes additional challenge to model DEMES. Considering dielectric elastomer and bending frame, respectively, as a hyper-elastic film and an elastic beam, Euler–Lagrange equation is employed to derive the equation of motion of actuator subject to a harmonic voltage. Fixed point analysis of the equation of motion demonstrates supercritical pitchfork bifurcation with respect to film pre-stretch as bifurcation parameter. It is shown by simulation results that the system possesses harmonic resonances as well as superharmonic and subharmonic resonance. Investigating Poincare map of the time response indicates that by changing excitation frequency and amplitude the response transforms from periodic to quasiperiodic and eventually aperiodic form.

Study on Vertical Vibration Characteristics of the 2-DOF Strip Rolling Mill Model with a Single Weak Defect on the Work Roll Bearing Outer Raceway

Considering the influence caused by a early single pit defect on the outer raceway of the work roll bearing, a 2-DOF plate strip rolling mill vertical vibration model with a single point weak fault on the outer raceway was established. With the practical parameters of the roughing mill of the 1780 hot continuous rolling mill, the vertical vibration characteristics of the rolling mill work roll with different rotating speed and different single pit defect area on the bearing outer raceway are analyzed by numerical simulation. It is found that with the change of the rotation speed of the work roll, different nonlinear vibration behaviors occurred, such as superharmonic resonance, main resonance, combined resonance and sub-harmonic resonance. Especially the subharmonic resonance of the work roll is more harmful than the main resonance when the work roll speed is twice the rotation speed corresponding to the first and second natural frequency of the rolling mill. This work provides a theoretical basis for further clarifying the effect caused by a early defect of the work roll bearing on the mill vibration.

Internal Resonance Phenomena in Coupled Ductile Cantilevers With Triple Frequency Ratio-Part II: A Mass Sensitivity Amplification Schemes

This paper, the second of two companion papers, reports a mass sensitivity amplification scheme, via amplitude and frequency shift enhancement of internal resonance (IR) in coupled ductile cantilevers with frequency ratio of three to one. The amplitude enhancement of super-harmonic resonance of order three in short cantilever can be achieved by the coupling of long cantilever via coupling overhang through the internal resonance between the low frequency mode and the high frequency mode. Such kind of enhancement mechanism is clarified by both experimental verification and numerical analysis. While the frequency shift enhancement of coupled cantilevers are experimentally demonstrated by comparatively analyzing the frequency shifts of single-cantilevers and coupled cantilevers with varying mass perturbations. The impact of coupling strength on frequency shifts of coupled cantilevers is also studied. The proposed scheme with coupled IR ductile cantilevers, can accomplish “sensing and detecting separation,” “amplitude and frequency shift enhancement,” and the linearity improvement of mass detecting, which is expected to be applicable to ultimate sensing of mass, force, field, and any other external perturbations, even under an atmospheric environments.

Primary and secondary resonance analyses of a cantilever beam carrying an intermediate lumped mass with time-delay feedback

Time-delay displacement and velocity feedback of different types of active control in a cantilever beam carrying an lumped mass is investigated in this paper. Based on Euler–Bernoulli beam theory, the nonlinear governing equation is studied with damping, harmonic distribution, displacement delay, velocity delay and two time delays. The multiple scales perturbation method is applied to obtain the frequency response equations near primary, superharmonic and subharmonic resonances. A thorough study on the stability is proposed, with a particular emphasis on delay feedback. The results show that the hardening and softening behaviors of the system depend on the location of lumped mass. Furthermore, the displacement feedback gain coefficient only makes the peak amplitude move to the low frequency, yet velocity feedback coefficient and their time delays can be used to effectively enhance the stability and quench the nonlinear vibration of the cantilever beam. Thus, reasonable selection of the control system parameters can effectively improve the level of vibration control for the mechanical system.

Efficient energy harvesting from nonlinear vibrations of PZT beam under simultaneous resonances

The present research aims to analyze a piezoelectric energy harvester under simultaneous effects of two hard excitations. The system consists of a unimorph cantilever beam carrying a tip mass; excited by two simultaneous hard base accelerations. The governing distributed-parameters partial differential equations (PDEs) of motion are obtained based on Euler-Bernoulli beam assumption accounting for geometric nonlinearity. Galerkin scheme is exploited to discretize the PDEs into a set of nonlinear ordinary differential equations (ODEs). The convergence analysis is carried out to obtain the minimum required number of modes. To generalize the further results, the dimensionless forms of ODEs are obtained. Then, the approximate-analytical solutions are derived based on multiple scales method (MSM) for the case of simultaneous occurrence of superharmonic and combination resonances. Frequency response curves of the displacement and voltage are plotted. Moreover, the harvester response is studied for the case of primary resonance. The results disclose that, the level of the generated voltage and power are amplified when the superharmonic and combinations resonances coexist in the system response. Furthermore, the system dynamics are investigated in the time domain, showing good agreement with the frequency domain findings.

A laboratory study of class III Bragg resonance of gravity surface waves by periodic beds

When moderately steep waves travel over a periodic rippled bed, class III Bragg resonance may occur due to the third-order quartet wave-bottom interaction among one bottom and three surface wave components. The theory, however, has not been experimentally confirmed. To verify the existence of class III Bragg resonance, we consider the simplest possible case involving a single incident wave and conduct a series of physical experiments. The experiments show that as the theory predicts, class III Bragg resonance could generate not only the reflected waves (from subharmonic resonance) but also the transmitted waves (from superharmonic resonance), and the reflection and transmission coefficients vary linearly along the rippled bottom patch. Furthermore, the experimental data of the reflected and transmitted waves (due to class III resonance) agree quantitatively well with the prediction by high-order spectral (HOS) method computations. To further understand the characteristics of class III resonance, we apply HOS simulations to study the more general cases including the presence of two incident waves of different frequencies as well as the presence of an incident wave group with Gaussian envelope. The results show that in addition to class I and class II Bragg resonances, class III Bragg resonance can significantly influence the evolution of surface wave fields passing over a rippled bottom, especially in the case of shallow water.

Size-dependent nonlinear forced vibration and dynamic stability of electrically actuated micro-plates

A size-dependent structural dynamic model that incorporates the effect of geometric nonlinearity is developed in this paper for the forced vibration and dynamic stability of thin rectangular micro-plates. The equations of motion for micro-plates are derived within the framework of classical plate theory, modified couple stress theory (MCST), and von Kármán geometric nonlinearity using Hamilton's principle. Galerkin method is used to convert the governing partial differential equations to a nonlinear second-order ordinary differential equation, which is solved by a Runge-Kutta method. The static instability analysis of the micro-plate is performed to determine the critical electrostatic voltages, and to avoid the pull-in instability. By tracking the static behavior of the microplate, and determining the electrostatic pull-in voltage, the frequency response curves are plotted. In dynamic response, primary, superharmonic, and subharmonic resonance are studied, and the frequency response equation is obtained for each case by the method of multiple scales. Further efforts are made to investigate the influence of size effect, electrical loading (DC and AC voltages), and excitation frequency on the static, and dynamic responses, critical AC voltages, and dynamic stability of micro-plates. It is found that the critical dynamic voltage is a function of the frequency of excitation force. It is shown that the stiffness of micro-plate decreases by increasing the constant DC voltage; however, the increase in the alternating AC voltage does not considerably affect the stiffness of the micro-plate.

Superharmonic Resonance of Fractional-Order Mathieu–Duffing Oscillator

The superharmonic resonance of fractional-order Mathieu–Duffing oscillator subjected to external harmonic excitation is investigated. Based on the Krylov–Bogolubov–Mitropolsky (KBM) asymptotic method, the approximate analytical solution for the third superharmonic resonance under parametric-forced joint resonance is obtained, where the unified expressions of the fractional-order term with fractional order from 0 to 2 are gained. The amplitude–frequency equation for steady-state solution and corresponding stability condition are also presented. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional-order term, excitation amplitudes, and nonlinear stiffness coefficient on the superharmonic resonance response of the system are analyzed in detail. The results show that the KBM method is effective to analyze dynamic response in a fractional-order Mathieu–Duffing system.

Earth body resonance

The full range of 72h-forced, 72 superharmonic resonance periods, is detected in time-series of all 866 earthquakes of (robust averages of) Mw5.6+ from USGS, EMSC, and GFZ, 2015-2019 catalogs. The resonance is in the 55’–15 days long-periodic band (0.303 mHz–0.771605 μHz) at 99–67% confidence. Moreover, omitting of the 21 overrepresenting events has improved the result. The signal is clear, strong, and stable – demonstrating beyond doubt that Mw6.2+ seismicity arises due to long-periodic resonance. Remarkably, the natural mode’s cluster was detected too; it averaged 60.1’, while the overall strongest resonance period was also 59.9’, at 2.3 var%, or to within the 1Hz sampling rate – revealing that the 72 h forcer is the modulator of the Earth’s natural period via synchronization. The dominance property of the forcer also follows from detection of its many other fractional multiples: 14/5, 3/2, 5/12, 5/36, etc. After Schumann resonance discovery in the short band (extremely long band of the EM Spectrum), this is the second report ever of a full resonance bundle in any global data, and the first ever in tectonic earthquakes occurrences. The Mw6.2+ seismotectonics arises via resonance-rupture response of tectonic plates and regions to the resonant phase or its fractional multiples. Fundamental questions of geophysics including earthquake prediction can be solved if the Earth is taken to be a multi-oscillator nonlinear system. As an immediate benefit, the find enables a reliable partial seismic anti-forecasting (prediction of seismic quiescence), months ahead globally. This discovery of mechanically induced extreme-band energy on Earth invalidates the main (heat-transfer) geophysical hypothesis and thus should drastically diminish the role of chemistry in geosciences, specifically of geochemistry.

Time-Delayed Feedback Control of Piezoelectric Elastic Beams under Superharmonic and Subharmonic Excitations

The time-delayed displacement feedback control is provided to restrain the superharmonic and subharmonic response of the elastic support beams. The nonlinear equations of the controlled elastic beam are obtained with the help of the Euler–Bernoulli beam principle and time-delayed feedback control strategy. Based on Galerkin method, the discrete nonlinear time-delayed equations are derived. Using the multiscale method, the first-order approximate solutions and stability conditions of three superharmonic and 1/3 subharmonic resonance response on controlled beams are derived. The influence of time-delayed parameters and control gain are obtained. The results show that the time-delayed displacement feedback control can effectively suppress the superharmonic and subharmonic resonance response. Selecting reasonably the time-delayed quantity and control gain can avoid the resonance region and unstable multi-solutions and improve the efficiency of the vibration control. Furthermore, with the purpose of suppressing the amplitude peak and governing the resonance stability, appropriate feedback gain and time delay are derived.

Nonlinearity in the Mechanical Response of Rubber as Investigated by High-Frequency DMA.

Nonlinear material response is analysed with the Fourier transform (FT) of the raw signal measured by a high-frequency dynamic mechanical analyzer (HF DMA). It is known from rheological behaviour of elastomers that reinforcing fillers additionally induce nonlinearity in an already inherently nonlinear system. This behaviour is often described in terms of a mechanical response of strain sweeps, essentially the transition from the linear viscoelastic (LVE) to the nonlinear viscoelastic (NVE) region. In the current investigation, the NVE region could be observed with respect to frequency under low-amplitude deformation. A foldover effect was observed, whereby the material exhibited a nonlinear dependency in relation to the increment of the filler amount above the percolation threshold. In addition, an apparent superharmonic resonance was observed within higher orders of vibrational modes which is further indication of nonlinearity. In this paper, the analytical approach is presented as a novel method to characterise the behaviour of the polymer–filler interaction by HF DMA.

Theoretical study of micro/nano-scale bistable plate for flexoelectric energy harvesting

Flexoelectricity offers a number of advantages in micro/nano-scale devices when compared with piezoelectricity as confirmed through numerous experiments and theoretical analyses. Snap-through behavior in bistable plate would convey a higher conversion efficiency for energy harvesting. However, there is no theoretical research on micro/nano-scale energy harvester with flexoelectricity based on bistable plate. This paper describes an electro-thermo-mechanical coupling system of micro/nano-scale bistable plates for piezoelectric energy-harvesting applications. The nonlinear geometric theory of Von Karman, strain gradient theory, the flexoelectric effect, and Hamilton’s principle were used to derive nonlinear dynamic formulas. The ambient vibration frequency and snap-through behavior both have an effect on the output voltages calculated by Matlab® software. An expression for the energy conversion efficiency of the bistable plate was derived and the influence of resistance on this efficiency was investigated. The result showed that the snap-through behavior delivered a large-amplitude vibration and a higher output voltage at the range of one-third super-harmonic resonance (2.8–3.5 MHz) for the micro/nano-scale bistable plate. This work will be helpful in the design of micro/nano-scale bistable plates with higher output voltages and improved conversion efficiencies for energy harvesting.

On the Application of the Multiple Scales Method on Electrostatically Actuated Resonators

We investigate modeling the dynamics of an electrostatically actuated resonator using the perturbation method of multiple time scales (MTS). First, we discuss two approaches to treat the nonlinear parallel-plate electrostatic force in the equation of motion and their impact on the application of MTS: expanding the force in Taylor series and multiplying both sides of the equation with the denominator of the forcing term. Considering a spring–mass–damper system excited electrostatically near primary resonance, it is concluded that, with consistent truncation of higher-order terms, both techniques yield same modulation equations. Then, we consider the problem of an electrostatically actuated resonator under simultaneous superharmonic and primary resonance excitation and derive a comprehensive analytical solution using MTS. The results of the analytical solution are compared against the numerical results obtained by long-time integration of the equation of motion. It is demonstrated that along with the direct excitation components at the excitation frequency and twice of that, higher-order parametric terms should also be included. Finally, the contributions of primary and superharmonic resonance toward the overall response of the resonator are examined.

The effectiveness of nonlinear integral positive position feedback control on a duffing oscillator system based on primary and super harmonic resonances

In this paper, we applied three different control methods; Positive Position Feedback (PPF), Integral Resonance Control (IRC) and Nonlinear Integrated Positive Position Feedback (NIPPF) added to a Duffing oscillator system subjected to harmonic force. An analytic solution is introduced using the multiple scales perturbation technique (MSPT) to solve the nonlinear differential equations, which simulate the system with NIPPF controller. Before and after control at the primary and superharmonic resonances, the nonlinear systems’ steady-state amplitude and stability are studied and examined. The influences of various parameters of the system after being connected to NIPPF are illustrated. Optimum working conditions for the NIPPF controller are obtained at internal resonance ratio 1:1. A Comparison is also made to validate the closeness between the numerical solution and the analytical perturbative one at time-history and frequency response curves (FRC). Finally, a comparison with the available results in the literature is presented. From this comparison, we find that the best control to the system is via the NIPPF controller.

Nonlinear analysis, instability and routes to chaos of a cracked rotating shaft

The aim of this paper is to explore the dynamical response of a rotating shaft with a cracked transverse section. This complex problem is distilled down to the study of a very comprehensive mechanical system composed of two non-cracked rigid bars connected with a nonlinear bending spring that concentrates the global stiffness of the cracked shaft. The switching crack model is presented as a specific case of the EDF–LMS family of models and adopted to describe the breathing mechanism of the crack. By analyzing the system equilibrium equations, we show that the stiffness change when the crack breathes leads to shocks with velocity jumps and coupling between the transverse oscillations. The linear stability of the periodic solutions is examined based on the Floquet’s theory. Nonlinear dynamics tools such as Poincare maps and bifurcation diagrams are used to unveil the system oscillations characteristics. Many well-known features due to the crack presence have been observed, but some unexpected responses are noticed like chaotic behavior. This can be confidently attributed to the abrupt change of the structure stiffness with the breathing crack. It has also been observed that with the superharmonic resonance phenomenon, the increase in static deflection accompanied by that of the first two harmonics amplitudes is a good indicator of a propagating crack presence.