Subharmonic Resonance Research Articles

Experimental investigation into the nonlinear dynamics of a bistable laminate

An experimental study of the single-well and twin-well, also referred to as intra-well and inter-well, respectively, nonlinear dynamics of a bistable composite laminate is presented. An asymmetric four-ply [0/90/0/90] carbon fiber laminate with two cylindrical stable equilibria supported at its center and free at all boundaries is used for the experimental testing. The mechanical bistability makes the laminate able to snap from one stable equilibrium to the other when displacement reaches a critical value. This snapthrough motion is highly nonlinear and associated with large-amplitude vibrations. This property opens chances for bistable laminates in morphing and energy-harvesting applications. An electromechanical shaker is used to excite the laminate at its center by a controlled amplitude and frequency excitation. The dynamic response of the laminate is measured using a Polytec laser vibrometer. In addition, four macro-fiber composite actuators are attached to the laminate to measure the output voltage due to vibration and hence assess its energy-harvesting capability. The laminate’s natural frequencies and damping under small-amplitude excitations that would match the natural frequencies of an underlying linear system are experimentally identified. A primary resonance excitation of the first bending mode is carried out using amplitude sweep and frequency sweep. It is shown that in both cases, the response starts as a small-amplitude single-well vibration around the static equilibrium position. Varying the excitation amplitude or frequency, the response exhibits a period-doubling bifurcation associated with higher levels of vibration. As the excitation conditions are varied further, the laminate snaps from one equilibrium position to the other. This snapthrough motion is characterized to be of a frequency that is half the natural frequency of the excited mode. It is shown that the period-doubling cascade is responsible for the escape from the single-well to the twin-well response. A secondary Hopf bifurcation that creates a period-three motion and a chaotic snapthrough were also identified. Two low-frequency interesting modes that are attributed to the elasticity of the midpoint support, referred to as rocking modes, were identified using a hammer. These modes got repeatedly excited via subharmonic resonance of order two. In terms of the energy harvesting, the snapthrough motion was found to greatly enhance the energy extraction capability. The frequency band of significant power generation was found to be about 25% of the excitation frequency, which is a good ratio for a single excitation frequency.

On the response of MEMS resonators under generic electrostatic loadings: experiments and applications

We present an investigation of the dynamic behavior of an electrostatically actuated clamped–clamped microbeam, under the simultaneous excitation of primary and subharmonic resonance. The simultaneous excitation of primary and subharmonic resonances of similar strength is experimentally investigated by using different combinations of AC and DC voltages. It is observed that the response of the resonator is governed by a mixed effect of both excitations. Subharmonic-dominated response shows sharp amplitude transitions and smaller monostable regime, while primary-dominated response shows gradual amplitude transition and larger monostable regime. Two potential applications are experimentally demonstrated. The first is a resonator-based MEMS AND logic gate based on AC only subharmonic excitation. The second is a charge sensor based on the transition from subharmonic-dominated response to primary-dominated response, which is potentially capable of detecting a small amount of electric charges.

Nonlinear vibrations of carbon fiber reinforced polymer laminated cylindrical shell under non-normal boundary conditions with 1:2 internal resonance

In this paper, the nonlinear vibrations of a carbon fiber reinforced polymer (CFRP) laminated cylindrical shell are investigated with 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance. The radial line load and axial excitation are acting on the two free ends of the CFRP laminated cylindrical shell. The partial differential governing equations of motion for the CFRP laminated cylindrical shell are derived by utilizing the von-Karman type nonlinear geometric relationship, the first-order shear deformation theory and Hamilton principle. Galerkin method is used to obtain two-degrees-of-freedom nonlinear ordinary differential governing equations of motion along the radial displacement of the CFRP laminated cylindrical shell under the non-normal boundary conditions. The ordinary differential governing equations are solved as a system of four-dimensional average equations by the second-order approximate multiple scale method. The first two order dimensionless natural frequencies versus the ratio of length to thickness L/h, the frequency-response curves, the force-response curves, the bifurcation diagrams, the three-dimensional bifurcation diagrams, the maximum Lyapunov exponent diagrams, the phase portraits, the time history diagrams and the Poincare maps are obtained by numerical calculations. The influences of the radial line load, the axial excitation as well as the detuning parameter of the CFRP laminated cylindrical shell on the 1:2 internal resonant behaviors are investigated.

Resonance Response of a Quasi-zero Stiffness Vibration Isolator Considering a Constant Force

Investigating the influence of additional constant force on the primary resonance and 1/3 sub-harmonic resonance of a quasi-zero stiffness vibration isolator. The frequency response curves (FRCs) of primary resonance and 1/3 sub-harmonic resonance are derived using the incremental harmonic balance method. The parametric analysis is then conducted to obtain the influence of additional constant force. First, increasing the constant force can lead to the increase of resonant frequency and make the system exhibit softening characteristic. The phenomenon of frequency overlap could occur in the FRC under certain conditions. Second, increasing the constant force can make the 1/3 sub-harmonic resonance region shrink. At last, the 1/3 sub-harmonic resonance can be avoided by controlling the excitation level within a certain small range.

Subharmonic resonance and critical eccentricity for the classical hydrogen atomic system

Subharmonic resonance behaviors are investigated for the classical hydrogen atom, with classical radiation damping and circularly polarized light acting on the classical electron. This study is intended for both potential experimental applications as well as for deeper theoretical purposes. Long resonant states are predicted for realistic Rydberg atoms and highly excited hydrogen states. Several previously undiscovered physical effects are predicted. First, the semimajor axis remains relatively constant when in subharmonic resonance; second, the eccentricity steadily increases until a maximum, critical value is reached, at which point orbital decay sets in. If the initial orbit is circular, this critical eccentricity value is shown to always be the same for each subharmonic condition, regardless of the initial orbital radius. An analytic derivation for this result is presented. The illustrated dynamics are of interest for the classical theory of stochastic electrodynamics (SED) regarding whether SED can fundamentally describe more of quantum phenomena, particularly atomic excited state behavior and related emission and absorption spectra. Also of interest are how classical resonances can be imposed on a near continuum of quantum states. Finally, there may be future technological applications, such as “reading” and “writing” information into Rydberg atoms in the form of subharmonic resonances.

Nonlinear dynamics of self-centring rocking steel frames using finite element models

Rocking post-tensioned steel frames capitalise on the use of rocking joints, and unbonded post-tensioning strands to provide self-centring action. Investigations on the complex and unconventional nonlinear dynamics of tied rocking steel frames, exclusive of supplemental damping methods, are presently limited. Increasing levels of energy-dissipation reduce the probability of observing nonlinear dynamic phenomena such as co-existing (high/low) amplitude responses at and around the system's nonlinear resonance. To this end, a finite element (FE) modelling framework is presented, validated and extended to multi-storey steel buildings. It is shown that the simulation strategies proposed enable an accurate representation of the complex nonlinear dynamics of self-centring structures, over a wide range of excitation frequencies and amplitudes. The methodology, applied to multi-storey steel frames, captures the presence of sub-harmonic resonances and higher-modes. It is also demonstrated that the additional demands observed in the rocking columns are the consequence of the asymmetry of the member boundary conditions.

Parametric Resonance of Pipes with Soft and Hard Segments Conveying Pulsating Fluids

In this paper, the parametric resonance of pipes with soft and hard segments induced by pulsating fluids is investigated. The lowest six natural frequencies and mode shapes of the soft–hard combination pipe simply supported at both ends are obtained by the modified Galerkin's method. The Floquet method is used to numerically determine the parametric resonance regions, including subharmonic resonance regions and combination resonance regions. The parametric resonance results are verified by comparison with published ones, which confirm the validity of the present model establishment and numerical calculation. Compared with a uniform pipe conveying fluid simply supported at both ends, the soft–hard pipe conveying fluid is found to reveal different dynamical behaviors. Decreasing the length of the soft pipe, while increasing the stiffness ratio of the hard pipe compared to the soft one, can effectively improve the stability of the pipe system. The parametric resonance results show that the mean flow velocity and pulsation amplitude of the fluid have a great influence on the width of the parametric resonance regions. It is advisable that the ratio (the soft pipe/the whole pipe) of the length may be designed to be 0.4–0.5 for a flexural rigidity ratio (the hard pipe/the soft pipe) of 2. As the stiffness ratio (the hard pipe/the soft pipe) increases beyond 26, the hard pipe may be regarded as a rigid pipe. The probability of parametric resonance occurrence will be smallest if the soft–hard combination pipe is supported in a clamped–pinned way. For certain application cases, the safety design length of the two pipes with different materials can be determined through numerical calculation.

Exploiting the subharmonic parametric resonances of a buckled beam for vibratory energy harvesting

The potential of harvesting vibratory energy via a bistable beam subjected to subharmonic parametric excitations is investigated. The vibrating structure is a buckled beam with two stable equilibria separated by a potential barrier. The beam is subjected to a superposition of a static axial load beyond its buckling load and a harmonic axial excitation whose frequency is around twice the frequency of the buckled beam’s first vibration mode. A macro-fiber composite patch is attached to one side of the beam to convert the strain energy resulting from the beam’s oscillation into electricity. The study considers two regimes of excitations: an amplitude sweep and a frequency sweep. In the first regime, the amplitude of excitation is quasi-statically varied while the excitation frequency is tuned at twice the natural frequency of the first vibration mode. In the second regime, the excitation frequency is swept forward and backward around the subharmonic resonant frequency while the amplitude of excitation is kept constant. A theoretical model which governs the electromechanical coupling of the transverse vibrations of the beam and the output voltage is used to monitor the response as the excitation parameters are changed. An experimental setup is also built and a series of tests is performed to validate the theoretical findings. It is shown that, depending on the amplitude and frequency of excitation, the harvester can perform small-amplitude periodic intra-well motion, intra- and inter-well chaotic motions, as well as periodic inter-well motions. Experimental results also show that, as compared to the classical linear resonance, utilizing the sub-harmonic resonance of a bistable energy harvesters can result in a broadband frequency response.

Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects

• Size-dependent nonlinear hard oscillation of nanobeams. • Surface free energy effects on the nonlinear secondary resonance of nanobeam with various end supports. • Size-dependent frequency-response of a nanobeam for the both of subharmonic and superharmonic excitations. • Size-dependent amplitude-response of a nanobeam for the both of subharmonic and superharmonic excitations. The main objective of this study is to predict both the subharmonic and superharmonic resonances of the nonlinear oscillation of nanobeams in the presence of surface free energy effects. To this purpose, Gurtin–Murdoch elasticity theory is adopted to the classical beam theory in order to consider the surface Lame constants , surface mass density, and residual surface stress within the differential equations of motion. The Galerkin method together with the method of multiple scales is utilized to investigate the size-dependent response of nanobeams under hard excitations corresponding to various boundary conditions. A parametric analysis is carried out to indicate the influence of the surface elastic parameters on the frequency-response as well as amplitude-response of the nonlinear secondary resonance including multiple vibration modes and interactions between them. It is seen that for the superharmonic excitation, except for the clamped–free boundary condition, the jump phenomenon is along the hardening direction, while in the clamped–free end supports, it is along the softening direction. Moreover, it is revealed that for the subharmonic excitation, within a specific range of the excitation amplitude, the nanobeam is excited, and this range shifts to lower external force by incorporating the surface free energy effects. It is found that in the case of superharmonic excitation, the value of the excitation frequency associated with the bifurcation point at the peak of the frequency-response curve increases by taking the surface free energy effect into consideration.

A Modified Newmark Scheme for Simulating Dynamical Behavior of MDOF Nonlinear Systems

The Newmark integration scheme, originally developed for simulating responses of linear dynamical systems, is modified in this paper to effectively model nonlinearities through introduction of the incremental displacements and the effective mass, damping and tiffness matrices. Based on the results of comparisons with the analytical solutions and the Runge-Kutta (RK) method for well-known nonlinear oscillators, the proposed scheme is found to capture accurately dynamical behaviors of nonlinear systems such as mplitude-dependent stiffening or softening natural frequencies, jump phenomena, superharmonic resonances, sub-harmonic resonances, and even chaos and bifurcations. The proposed scheme is more efficient than the RK method for large scale nonlinear ystems with some type of sparsity for which a targeted algebraic equations solver can be employed to speed up the solution for each time step.

Coexistence of Subharmonic Resonant Modes Obeying a Period-Adding Rule

We report numerical evidence of a novel chain of subharmonic resonances organized in period-adding domains in the parameter space. Different periodic domains are mediated by a coexisting main subharmonic motion according to a hierarchical organization rule. We consider the driven bilinear oscillator — which is the simplest system to model a range of engineering systems going from cracked mechanical structures to switching electronic circuits — and show that its parameter space is marked by multistability involving subharmonic main modes, as well as period-adding secondary modes which, through a period-doubling route, lead to coexisting chaotic attractors.

Stability research of multistage gear transmission system with crack fault

The sub-harmonic resonance phenomenon exists in multistage gear transmission system. Collision motion leads the sub-harmonic resonance instability, which brings harm to system stability. So investigating instability conditions of sub-harmonic resonance and effects of fault on stability have important significance on controlling system stability. From the point of view of nonlinear dynamics, dimensionless dynamical equations of multistage gear transmission system which contains a two-stage fixed-axis gear and a one-stage planetary gear are established. The bifurcation diagrams of system are obtained. By the changing of Poincaré section analyze the evolution process of sub-harmonic resonance and instability conditions. The changes of sub-harmonic resonance and system stability are studied under fixed-axis gear tooth crack fault state in different clearance. The study has important significance for operational prediction of system.

Synchronisation vs. resonance: Isolated resonances in damped nonlinear oscillators

We describe differences between synchronisation and resonance, and analyse different types of nonlinear resonances in a weakly damped Duffing oscillator using bifurcation theory techniques. In addition to previously reported (i) odd subharmonic resonances found on the primary branch of symmetric periodic solutions with the forcing frequency and (ii) even subharmonic resonances due to symmetry-broken periodic solutions that bifurcate off the primary branch and also oscillate at the forcing frequency, we uncover (iii) novel resonance type due to isolas of periodic solutions that are not connected to the primary branch. These occur between odd and even resonances, oscillate at a fraction of the forcing frequency, and give rise to a complicated resonance ‘curve’ with disconnected elements and high degree of multistability.We use bifurcation continuation to compute resonance tongues in the plane of the forcing frequency vs. the forcing amplitude for different but fixed values of the damping rate. Our analysis shows that identified here isolated resonances explain the intriguing “intermingled tongues” that were observed for weak damping and misinterpreted as (synchronisation) Arnold tongues in Paar and Pavin (1998). What is more, isolated resonances link “intermingled tongues” to a seemingly unrelated phenomenon of “bifurcation superstructure” described for moderate damping in Parlitz and Lauterborn (1985).

Analytical Modeling and Experimental Verification of Nonlinear Mode Coupling in a Decoupled Tuning Fork Microresonator

This paper provides the analytical and experimental studies into the nonlinear mode coupling in a tuning fork microresonator, with electrostatic actuation at forced and internal resonance frequencies. The analytical investigation leads to modal equations that take into account kinematic and electrostatic nonlinearities. The equations of motion are derived using Lagrange’s energy method. The theoretical resonance curves are determined by using the two-variable expansion perturbation technique near simultaneous subharmonic and internal resonances. The two desired mode shapes of the device are tailored to establish an approximate two-to-one frequency ratio between their natural frequencies. A sample device design is fabricated in a commercial foundry process. The influence of AC driving levels and a detuning parameter, due to inevitable fabrication imperfections, on the system dynamics are investigated theoretically and experimentally. It is shown that the deviation from ideal internal resonance condition separates the region of the nonlinear vibration into two distinct excitation frequencies. The model and simulated results obtained by the perturbation analysis are validated by comparing them with the experimental results. [2017–0297]

Moment Lyapunov Exponent and Stochastic Stability of Binary Airfoil under Combined Harmonic and Non-Gaussian Colored Noise Excitations

Both periodic loading and random forces commonly co-exist in real engineering applications. However, the dynamic behavior, especially dynamic stability of systems under parametric periodic and random excitations has been reported little in the literature. In this study, the moment Lyapunov exponent and stochastic stability of binary airfoil under combined harmonic and non–Gaussian colored noise excitations are investigated. The noise is simplified to an Ornstein-Uhlenbeck process by applying the path-integral method. Via the singular perturbation method, the second-order expansions of the moment Lyapunov exponent are obtained, which agree well with the results obtained by the Monte Carlo simulation. Finally, the effects of the noise and parametric resonance (such as subharmonic resonance and combination additive resonance) on the stochastic stability of the binary airfoil system are discussed.

Determination of the Vibration Diagnostic Parameters Indicating the Presence of a Mode I Crack in a Blade Airfoil at the Main, Super- and Subharmonic Resonances

The paper presents the results of numerical calculations to determine the regularities underlying the influence of the characteristics of a mode I breathing fatigue crack on the leading edge of the aircraft gas turbine engine blade airfoil – that is responsible for the nonlinearity of the vibratory system – on its flexural forced vibration behavior. Their comparison with similar data for the rod of a rectangular cross section has shown satisfactory agreement. The calculation of the vibration diagnostic parameters indicating the presence of damage is performed using the developed finite element models of the investigation objects whose forced vibrations were excited by the kinematic displacement of fixed edge elements. The breathing crack is modeled as a mathematical cross section, the non-penetration of its faces is ensured by solving the contact problem of their interaction. The problem of the system forced vibrations is solved using the Newmark method and fast Fourier transformation. The obtained amplitude-frequency characteristics of the undamaged and damaged blade airfoil are indicative of the increase in the energy dissipation in the system in the presence of a breathing crack. The change in the resonant vibration frequency of the cracked object of investigation and the amplitude ratio of the dominant harmonics of displacements and accelerations (the ratio of the former to the latter at the superharmonic resonance and the latter to the former at the main and subharmonic resonances) were chosen as the vibration diagnostic indicators of damage. The amplitude ratio of the dominant harmonics of accelerations at the superharmonic resonance and that of displacements at the subharmonic resonance is found to be the most sensitive indicator of the presence of cracks.

Computer Simulation of High-Frequency Heating of a Protonated Poly(ethylene oxide) Chain in a Vacuum

Heating of an isolated protonated poly(ethylene oxide) chain by high-frequency electric field was simulated using molecular dynamics. Simulations were performed at the field strength of about 108 V/m and frequencies from 1 to 100 GHz. At initial temperature (500 K), the chain is a globule. Absorption of the field energy heats the chain and leads to a growth in its size. In some frequency ranges, the field causes the chain to rotate as a whole. The rotation leads to a sharp increase in the heating rate and the rate of stretching the chain. The rotating chain has larger size than its non rotating counterpart at the same temperature. Contrary to the non rotating chain, this chain can be in a two-phase state in which the globular and the stretched microphases coexist. The absorption spectrum of the chain is temperature dependent and, at high temperatures, contains two distinctly expressed peaks. In the vicinity of these peaks, the chain rotates with a frequency equal to the field frequency or about half the field frequency. The results obtained are explained using the model of a spatial damped oscillator in which, together with the usual resonance, the subharmonic resonances typical for nonlinear systems arise.

A parametric electrostatic resonator using repulsive force

In this paper, parametric excitation of a repulsive force electrostatic resonator is studied. A theoretical model is developed and validated by experimental data. A correspondence of the model to Mathieu's Equation is made to prove the existence and location of parametric resonance. The repulsive force creates a combined response that shows parametric and subharmonic resonance when driven at twice its natural frequency. The resonator can achieve large amplitudes of almost 24 μm and can remain dynamically stable while tapping on the electrode. Because the pull-in instability is eliminated, the beam bounces off after impact instead of sticking to the electrode. This creates larger, stable trajectories that would not be possible with traditional electrostatic actuation. A large dynamic range is attractive for MEMS resonators that require a large signal-to-noise ratio.

Novel threshold pressure sensors based on nonlinear dynamics of MEMS resonators

Triggering an alarm in a car for low air-pressure in the tire or tripping an HVAC compressor if the refrigerant pressure is lower than a threshold value are examples for applications where measuring the amount of pressure is not as important as determining if the pressure has exceeded a threshold value for an action to occur. Unfortunately, current technology still relies on analog pressure sensors to perform this functionality by adding a complex interface (extra circuitry, controllers, and/or decision units). In this paper, we demonstrate two new smart tunable-threshold pressure switch concepts that can reduce the complexity of a threshold pressure sensor. The first concept is based on the nonlinear subharmonic resonance of a straight double cantilever microbeam with a proof mass and the other concept is based on the snap-through bi-stability of a clamped-clamped MEMS shallow arch. In both designs, the sensor operation concept is simple. Any actuation performed at a certain pressure lower than a threshold value will activate a nonlinear dynamic behavior (subharmonic resonance or snap-through bi-stability) yielding a large output that would be interpreted as a logic value of ONE, or ON. Once the pressure exceeds the threshold value, the nonlinear response ceases to exist, yielding a small output that would be interpreted as a logic value of ZERO, or OFF. A lumped, single degree of freedom model for the double cantilever beam, that is validated using experimental data, and a continuous beam model for the arch beam, are used to simulate the operation range of the proposed sensors by identifying the relationship between the excitation signal and the critical cut-off pressure.

Subharmonic resonances in high-order wave mixing in the quantized atomic motion in a one-dimensional optical lattice

We report on the observation of subharmonic resonances in high-order wave mixing associated with the quantized vibrational levels of atoms trapped in a one-dimensional optical lattice created by two intense nearly counterpropagating coupling beams. These subharmonic resonances, occurring at $\ifmmode\pm\else\textpm\fi{}1/2$ and $\ifmmode\pm\else\textpm\fi{}1/3$ of the frequency separation between adjacent vibrational levels, are observed through phase-match angularly resolved six- and eight-wave mixing processes. We investigate how these resonances evolve with the intensity of the incident probe beam, which couples with one of the coupling beams to create anharmonic coherence gratings between adjacent vibrational levels. Our experimental results also show evidence of high-order processes associated with coherence involving nonadjacent vibrational levels. Moreover, we also demonstrate that these induced high-order coherences can be stored in the medium and the associated optical information retrieved after a controlled storage time.