Subharmonic Resonance Research Articles

Secondary Resonance Energy Harvesting with Quadratic Nonlinearity

Piezoelectric energy harvesters can transform the mechanical strain into electrical energy. The microelectromechanical transformation device is often composed of piezoelectric cantilevers and has been largely experimented. Most resonances have been developed to harvest nonlinear vibratory energy except for combination resonances. This paper is to analyze several secondary resonances of a cantilever-type piezoelectric energy harvester with a tip magnet. The conventional Galerkin method is improved to truncate the continuous model, an integro-partial differential equation with time-dependent boundary conditions. Then, more resonances on higher-order vibration modes can be obtained. The stable steady-state response is formulated approximately but analytically for the first two subharmonic and combination resonances. The instability boundaries are discussed for these secondary resonances from quadratic nonlinearity. A small damping and a large excitation readily result in an unstable response, including the period-doubling and quasiperiodic motions that can be employed to enhance the voltage output around a wider band of working frequency. Runge–Kutta method is employed to numerically compute the time history for stable and unstable motions. The stable steady-state responses from two different methods agree well with each other. The outcome enriches structural dynamic theory on nonlinear vibration.

Nonlinear resonances phenomena in a modified Josephson junction model

In this paper, the equivalent circuit of the non-autonomous Josephson junction (JJ) is presented and the effect of the proper frequency on the phase ϕ is studied. We also study nonlinear resonance phenomena in the oscillations of a modified Josephson junction (MJJ). These oscillations are probed through a system of nonlinear differential equations and the multiple time scale method is employed to investigate all different types of resonance that occur. The results of primary, superharmonic and subharmonic resonances are obtained analytically. We show that the system exhibits hardening and softening behaviors, as well as hysteresis and amplitude hopping phenomena in primary and superharmonic resonances, and only the hysteresis phenomenon in subharmonic resonance. In addition, the stabilities and the steady state solutions in each type of resonances are kindly evaluated. The number of equilibrium points that evolve with time and their stabilities are also studied. Finally, the equations of motion are numerically integrated to check the correctness of analytical calculations. We further show that the dynamics of the MJJ is strongly influenced by its parameters.

Feedforward attractor targeting for non-linear oscillators using a dual-frequency driving technique.

A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.

Nonlinear Parametric Vibration of the Geometrically Imperfect Pipe Conveying Pulsating Fluid

In this paper, the novel model of fluid-conveying imperfect pipe supported at both ends is established by considering the geometric imperfection and the geometric nonlinearity induced by mid-plane stretching. The integral-partial differential equation is discretized by the Galerkin method and solved by a fourth-order Runge–Kutta integration algorithm. Compared with the supercritical pitchfork bifurcation of the perfect pipe conveying fluid, the results show that the cusp bifurcation occurs in the imperfect pipe when increasing the flow velocity. Excellent agreement is observed between the numerical results and the analytical results. The two stable asymmetry bifurcation branches bring interesting phenomena in the post-buckling state. The global nonlinear dynamic behaviors of the imperfect pipe are studied by establishing the bifurcation diagrams. The influence of the geometric imperfection amplitude on the nonlinear response is leading to cusp bifurcation comparing with pitchfork bifurcation of the perfect pipe. When pulsation frequency is set as the bifurcation parameter, there are clear nonresonance ranges, low energy resonance ranges and high energy resonance ranges. In the high energy resonance ranges, the first mode vibration coexisting with the sub-harmonic resonance and combination resonance occurs. As the mean flow velocity and pulsation amplitude are set as bifurcation parameters, the vibration of the imperfect pipe becomes more and more complicated. The vibration exhibits far richer dynamic behaviors including periodic, multi-periodic, quasi-periodic, and chaotic motions. The viscoelastic damping can effectively suppress the vibration response and transfer the high energy resonance to the low energy resonance state. The improved model and corresponding results provide useful information for further studying the dynamic behaviors of fluid-conveying pipe with geometric imperfections.

Bifurcations in a Pre-Stressed, Harmonically Excited, Vibro-Impact Oscillator at Subharmonic Resonances

This study investigates the behavior of a damped, inelastic, sinusoidally forced impact oscillator which has its barrier placed such that the oscillator always vibrates under compression about its subharmonic resonant frequencies. The Poincaré sections at near subharmonic resonance conditions exhibit finger-shaped chaotic attractors, similar to the strange attractor mapping of Hénon and the ones found by Holmes in his study of chaotic resonances of a buckled beam. The number of such fingers are observed to increase as the barrier distance from the equilibrium is decreased. These chaotic states are interspersed with regimes of periodic behavior, with the periodicity being in accordance with well defined period adding laws. This study also focuses on the ordered behavior of the one-impact period-[Formula: see text] orbits around the higher subharmonics of the oscillator.

Nonlinear superharmonic and subharmonic resonances of piezoelectric elliptic thin plates under thermoelastic coupling effect

This paper aims at investigating the nonlinear vibration characteristics of the piezoelectric elliptic thin plate under the combined action of external harmonic excitation and temperature field. Based on the Von-Karman plate theory of large deflection, the nonlinear governing equation is derived by using the Bubnov-Galerkin method. Then, the amplitude-frequency response equations are further obtained using the multi-scale method, and the stability conditions of the steady solution are determined according to the Routh-Hurwitz criterion. The effects of the temperature difference at the plate center, the effective damping coefficient, the external excitation amplitude, and the ratio of the semi-major axis to the semi-minor axis on superharmonic and subharmonic resonances behaviors of the piezoelectric elliptic thin plate are analyzed. It is found that the coexistence region of multiple solutions and the resonant amplitude decrease for superharmonic resonance, as the temperature difference at the plate center increases. For subharmonic resonance, with the increase in the temperature difference at the plate center, the response amplitude increases, while the distance between two branches of the amplitude-frequency response curve has no obvious change. The softened/hardened nonlinear characteristics of the system are determined by the ratio of the semi-major axis to the semi-minor axis of the piezoelectric elliptic thin plate.

Influences of generic payload and constraint force on modal analysis and dynamic responses of flexible manipulator

This article presents the dynamic modeling and nonlinear analysis of two-link flexible manipulator with generic payload whose center of gravity is different than the point of attachment with the link. The manipulator is driven by harmonic joint motions and subjected to a constraint force at the terminal end due to its interaction with working environment. Hamilton’s approach is adopted to dynamically model the multi-link manipulator for longitudinal and transverse vibrations incorporating the nonlinearities associated with axial stretching. The resulting set of eigenfrequency equations are solved numerically with subsequent generation of eigenspectrums. Further, this paper investigates the resonant behavior of manipulator links for the subharmonic and primary resonance conditions due to end follower force and revolute joint motions, respectively. The resonant responses show the jump phenomenon and existence of multi-valued solutions at pitchfork and saddle-node bifurcation points. The numerical and graphical results thus obtained, clearly indicate the significant influence of variation of system configurations and generic payload on eigencharacteristics and nonlinear responses, and hence must be given consideration during dynamic modeling of a manipulator in order to attenuate undesirable vibrations.

Efficiency of mono-stable piezoelectric Duffing energy harvester in the secondary resonances by averaging method. Part 1: Sub-harmonic resonance

The smart structures with the harmonic nonlinear vibration are used widely to broaden the frequency range of vibrations energy harvesting. In the paper, the potential of harvesting vibratory energy via a mono-stable Duffing energy harvester subjected to sub-harmonic excitation is investigated. The averaging method is used to derive the approximate solutions of the electromechanical linkage management equation system. The output voltage, power and energy harvesting efficiency of the harvester are investigated.

Energy harvesting from the secondary resonances of a nonlinear piezoelectric beam under hard harmonic excitation

This paper investigates the dynamical response of a nonlinear piezoelectric energy harvester under a hard harmonic excitation and assesses its output power. The system is composed of a unimorph cantilever beam with a tip mass and exposed to an harmonic tip excitation with a hard forcing amplitude. First, the governing dimensionless nonlinear electromechanical ordinary differential equations (ODEs) are obtained. Next, the multiple scales method (MSM) is exploited to provide an approximate-analytical solution for the ODEs in hard and soft forcing scenarios. It is observed that, the hard force results in sub- and super-harmonic resonances. The MSM-based solutions are then validated by a numerical integration method and a good agreement is observed between the approximate-analytical and numerical results. Furthermore, utilizing the MSM-based solutions for the subharmonic, superharmonic, and soft primary resonances cases, the associated frequency and force response curves are constructed. It is revealed that the hard excitation leads to a remarkable voltage generation in the secondary resonances; this leads to a broadband energy harvesting. In addition, the time-domain electrical responses of the secondary resonances are also obtained and compared with each other. Finally, the three-dimensional graphs of the electrical power versus detuning parameter and time constant ratio in the cases of the secondary resonances are plotted. The results show that the optimum output power of the superharmonic resonance is considerably larger than the maximum power of the subharmonic resonance case.

Vortex-induced vibrations of a circular cylinder with two symmetrically arranged control rods

In this paper, the vortex-induced vibration (VIV) of a main cylinder with two symmetrically arranged control rods at Reynolds number of 500 is numerically simulated. The control rods and the main cylinder are connected rigidly and can move synchronously, which is modeled as a spring oscillator with double degree of freedom. Seven gap ratios (G/D= 0.1 ∼2) and sixteen angles (α= 15° ∼165°) are used to model the cylinder system. Thirteen flow patterns are identified, and the VIV of the cylinder system in every flow pattern is discussed separately. It is found that the characteristics of the VIV of the cylinder system are similar in the same flow pattern, which may give a way to understand the VIV of the multiple cylinders further. Based on the classification in previous researches, some advanced insights into the passive control are achieved. The control rods located in the boundary layer (G/D= 0.1) can change the VIV of the main cylinder most effectively, and the effect decreases with increasing gap ratio. The control rods immerged in the wake of the main cylinder may result in sub-harmonic resonance, while the control rods located centrally in front of the main cylinder does not obviously affect the VIV of the main cylinder. As G/D= 0.1 ∼0.7, overall, the control rods located nearly side-by-side with main cylinder can amplify the VIV, while the control rods located downstream of the main cylinder can suppress the VIV. As G/D= 1 ∼2, the control rods can suppress the VIV slightly, and the effect increases with increasing gap ratio and decreasing angle. For G/D= 0.1 & α= 105°, the VIV is magnified for maximum effect, and the peak amplitude is about twice that of the single cylinder. For G/D= 0.2 & α= 45°, the VIV is suppressed for maximum effect, and the peak amplitude is about 0.88% of that of the single cylinder.

Asymmetric Large Deformation Superharmonic and Subharmonic Resonances of Spiral Stiffened Imperfect FG Cylindrical Shells Resting on Generalized Nonlinear Viscoelastic Foundations

This paper is devoted to superharmonic and subharmonic behavior investigation of imperfect functionally graded (FG) cylindrical shells with external FG spiral stiffeners under large amplitude excitations. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler–Pasternak foundation augmented by a Kelvin–Voigt viscoelastic model with a nonlinear cubic stiffness, to account for the vibration hardening/softening phenomena and damping considerations. The von Kármán strain-displacement kinematic nonlinearity is employed in the constitutive laws of the shell and stiffeners. The external spiral stiffeners of the cylindrical shell are modeled according to the smeared stiffener technique. The coupled governing equations are solved by using Galerkin’s method in conjunction with the stress function concept. The multiple scales method is utilized to detect the subharmonic and superharmonic resonances and the frequency–amplitude relations of the 1/3 and 1/2 subharmonic and 3/1 and 2/1 superharmonic resonances of the system. Finally, the influences of the stiffeners helical angles, foundation type, coefficient of the nonlinear elastic foundation, material distribution, and excitation amplitude on the system resonances are investigated comprehensively.

Nonlinear Vibration Mitigation of a Beam Excited by Moving Load with Time-Delayed Velocity and Acceleration Feedback

The time-delayed velocity and acceleration feedback control are provided to mitigate the resonances response of a nonlinear dynamic beam. By use of the method of multiple scales, the primary resonance and the 1/3 subharmonic resonance response of the controlled beam are analyzed. The excitation amplitude response peak and critical expression are obtained, and numerical simulations are also given. The effect of the feedback gains and time delayed on the steady-state response of the two types of resonances are investigated. The result show that time-delayed acceleration feedback control can effectively mitigate amplitude, and the main resonance response is affected periodically. Selecting reasonable control gain and time delay quantity can avoid the main resonance region and unstable multi-solutions, and can improve the efficiency of the vibration control.

Subharmonic and Combination Resonance of Rotating Pre-deformed Blades Subjected to High Gas Pressure

The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance, including two subharmonic resonances and two combination resonances. The dimensionless gas excitation amplitude is assumed to share the same order with the dimensionless vibration displacement. Four cases of resonance are confirmed by examining the secular terms. The theoretical analysis framework is established for each resonance case based on the method of multiple scales. The original dynamic system is integrated numerically by the Runge–Kutta method. The frequency components and phases obtained from fast Fourier transform of the numerical response are used to verify the theoretical results. For the purpose of contrast, modulation equations are also integrated numerically. In all four resonance cases, the theoretical results agree well with the numerical simulation. Parameter studies are conducted to clarify the effects of system parameters on the perturbation curves. Various results are obtained for the rotating blade. A quasi-saturation phenomenon occurs in both combination resonances of summed type and difference type, and the corresponding limit value of the second-mode response can be reduced by decreasing the external detuning parameter. The quasi-saturation phenomenon of rotating blade only appears with high gas pressure. The subharmonic resonance of second mode and the combination resonance of summed type are hard to excite in practice compared with the other two cases.

Nonlinear vibrations of fluid-conveying FG cylindrical shells with piezoelectric actuator layer and subjected to external and piezoelectric parametric excitations

Nonlinear vibrations of a fluid-conveying functionally graded (FG) cylindrical shell with piezoelectric actuator layer and subjected to external excitation and piezoelectric parametric excitation are analyzed theoretically. Considering the electric-thermo-fluid-structure interaction effect, a nonlinear dynamic model of fluid-conveying FG cylindrical shell with piezoelectric actuator layer is developed based on Hamilton’s principle and von-Karman geometrical nonlinearity. The inviscid, incompressible, isentropic and irrotational fluid is coupled into governing equations using the linearized potential theory. The nonlinear coupled differential governing equations of system are obtained by using Galerkin’s method with two modes. The developed coupled model is validated by comparing with the prior data and good agreements are observed. Multiple scales method is used to obtain nonlinear dynamic behaviors of the coupled system in case of 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance. The effects of three coupling cases between two modes on the dynamic behaviors of system are explored. Considering strongly coupled effect between two modes, the effect of external excitation, damping coefficient, piezoelectric harmonic voltage, fluid flow velocity, temperature and volume fraction exponent of FG cylindrical shell on the frequency-response curves are investigated. The influence of detuning parameters on force-amplitude response and voltage-amplitude response of system are also discussed.

Aeroelastic responses of airfoil under dynamic stall forced to oscillate by cyclic pitch input

The helicopter blades are involved in a nonlinear aeroelastic system in which unsteady dynamic stall flow and flexible structure are coupled. This problem can be reduced to a 2D aeroelastic problem in which airfoil is forced to oscillate through cyclic pitch input. In this paper, a modified dynamic stall model which focuses on revising the modeling of aerodynamic loads in reattachment stage is adopted in the numerical simulation to predict the aeroelastic responses of the above-mentioned system. The results are validated against experiments, after which parametric studies are performed to investigate the complex aeroelastic responses. In single Degree-Of-Freedom (DOF) system, secondary resonances happen in specific ratios of natural frequency to forcing frequency, which are found in both experiments and simulation. Moreover, as free-stream velocity varies, the aerodynamic stiffness has an apparent influence on the pitching natural frequency of system with low structural frequency. The varying pitching natural frequency brings about different aeroelastic responses including 1/2 subharmonic resonances and quenching phenomenon in single-DOF system as well as internal resonances and combination resonances in 2-DOF system. The aeroelastic system with low structural frequency is further investigated through multiple scale method, which provides an illustration for the mechanism of those phenomena found in the aeroelastic simulation adopting a semi-empirical aerodynamic model.

Nonlinear resonant behavior of thick multilayered nanoplates via nonlocal strain gradient elasticity theory

In this study, the nonlinear forced vibration of simply supported multilayered nanoplates in the framework of the first-order shear deformation of Mindlin plate theory based on the nonlocal strain gradient elasticity theory is investigated. The nonlinear von Karman strain–displacement relation is employed. The interaction of van der Waals forces between adjacent layers of a multilayered nanoplate is considered. The harmonic balance method is used to analyze the nonlinear resonance behavior of a thick nanoplate. The primary resonance and the secondary resonance which consist of superhamonic and subharmonic resonance are investigated. As a main result, by increasing the thickness of a thick nanoplate, the subharmonic resonance disappears and the frequency response curves imply non-resonant behavior, responses with finite amplitudes and different shapes for different thicknesses of thick nanoplates. At the end, the natural frequencies of thick multilayered nanoplates are obtained, and the effect of variation of nonlocal parameter, strain gradient parameter, number of layers and thickness of the nanoplate on the frequency response curves and stable and unstable regions is investigated.

Investigation of the 1/2 order subharmonic emissions of the period-2 oscillations of an ultrasonically excited bubble

In this work, through applying a comprehensive bifurcation analysis method, we study the nonlinear radial oscillations of the bubble oscillator. The frequency of the driving force is chosen as the linear resonance frequency (fr) and linear subharmonic (SH) resonance frequency (fsh=2fr) of the bubble. Results show that, when the bubble is sonicated with 2fr, period doubling is more likely to result in non-destructive oscillations. The evolution of the period-2 (P2) oscillations of the bubble makes the shape of a bowtie for bubbles with an initial diameter of 0.74 μm and above. When f=2fr, the phase portrait of the P2 attractor is distinctly different from a P2 attractor when f=fr, and subharmonic component of the backscattered pressure is maximum. When sonicated by 2fr, due to lower oscillation amplitude and gentler bubble collapse, the bubble can sustain stable P2 oscillations for a longer duration and over a broader range of applied acoustic pressure.

Active vibration control of truncated conical shell under harmonic excitation using piezoelectric actuator

Abstract In this study, active vibration control of an FGM truncated conical shell, subjected to the harmonic excitation, is investigated via a semi-analytical method. The governing equations of truncated conical shells in conjunction with piezoelectric layers are derived using Donnell shell theory and von Karman's non-linearity. Then, the dynamic equation of the system is discretized using Galerkin method to obtain the ordinary differential equation of system. Using the semi-analytical method by means of perturbation theory, the frequency response of superharmonic and sub-harmonic analysis of the system is performed. Then, using the piezoelectric smart materials, the active vibration control for the truncated conical shell is investigated and frequency-amplitude functions are analyzed for the superharmonic and subharmonic resonances. The results presents the influence of excitation amplitude and cone vertex angle on the frequency-amplitude response and in specific the effect of different control gain coefficients on the system's vibration behavior. Finally, the performance of active control strategy is investigated by means of the time response of system equipped with piezoelectric layers for different types of the FG distribution in thickness direction.

The homotopy analysis for the nonlinear dynamics of planetary gear trains

In this paper, the nonlinear oscillation of planetary gear trains is investigated by the homotopy analysis method. The nonlinearity of planetary gear trains due to the periodically time-varying mesh stiffness and contact loss are included. In contrast to the perturbation analysis, the homotopy analysis method is independent of the contact loss ratio, and then can be applied to both small and large contact loss ratios. In this article, firstly the closed-form approximations for the primary resonance, sub-harmonic resonance, and super-harmonic resonance are obtained by homotopy analysis method. The accuracy of homotopy analysis method solutions is evaluated by numerical integration simulations. Results indicate that with relatively large contact loss ratios, the amplitude–frequency curves obtained by homotopy analysis method agree better with the results obtained by numerical integration than those obtained by the method of multiple scales. This study lays a higher accurate foundation for more complex nonlinear dynamic analysis of planetary gear trains.

PRIMARY AND SUBHARMONIC SIMULTANEOUS RESONANCE OF DUFFING OSCILLATOR 1)

In this paper, the dynamics and stability of the Duffing oscillator subjected to the primary resonance together with the 1/3 subharmonic resonance are studied. At first, the approximate analytical solution and amplitude-frequency equation are obtained through the method of multiple scales, and the correctness and satisfactory precision of the approximate solution are verified by simulation. Then, the amplitude-frequency equation and phase-frequency equation of steady-state response are derived from the approximate analytical solution, and it can be found there are at most seven different periodic solutions, which are called multi-value characteristics and can be used to switch the state of the system. Moreover, the stability condition of steady-state response is derived based on Lyapunov theory, and the amplitude-frequency curves of steady-state response are compared with the cases where the primary or 1/3 subharmonic resonance exists alone, and it is found that the system contains both resonance characteristics. At last, the effects of nonlinear factor and excitations on the system response are analyzed by simulation. The particular phenomena in this system are revealed, i.e., the nonlinear factor affects the response amplitude, multi-value characteristics and stability of the system with stiffness softening. However, for the stiffness hardening system, the nonlinear factor only affects the response amplitude, which is similar to the cases of single-frequency excitation. These results are important for the study on the Duffing system or other similar systems.