Internal Resonance Phenomena Research Articles

Perturbation analysis of the two-to-one internal resonance of planetary gear trains

In this study, the two-to-one internal resonance between the first two rotational modes of planetary gear trains (PGTs) is investigated. A purely rotational model is applied considering mesh stiffness variations, tooth separations, and tooth profile modifications (TPMs). Semi-analytical solutions for the internal resonance case are obtained using the method of multiple scales (MMS). The solution equations indicate that the mesh stiffness variations and tooth separations are the main factors causing internal resonance. A validation of the MMS was performed by numerical integration (NI). The results from an example analysis indicate that there exists an internal resonance phenomenon in the case of ωN+2 ≈ ω2, where ω2 and ωN+2 are the natural frequencies associated with the rotational modes, and N is the number of planet gears. Internal resonance in PGTs causes chaos, and part of the energy is transmitted from the ring gear to the sun gear through shocks. Proper TPMs that eliminate the tooth separations could suppress the internal resonance. The internal resonance, in turn, affects the optimal areas of the TPM magnitudes.

Internal resonance in coupled oscillators – Part II: A synchronous sensing scheme for both mass perturbation and driving force with duffing nonlinearity

This paper, the second of two companion papers, mainly reports a synchronous sensing scheme for both mass perturbation and driving force, via the internal resonance phenomena in various coupled Duffing oscillators. The mass perturbation, applied on the low frequency oscillator, is sensed by the dip frequency of the low frequency oscillator caused by modal interaction, and further detected by the corresponding multiplied response frequency of the high frequency oscillator, while the driving force, applied on the low frequency oscillator, is directly detected by measuring the multiplied jump frequency of the high frequency oscillator. A magnetically coupled orthogonal beams with a frequency ratio of two to one, is taken as an example for both theoretical and experimental demonstration. The results show that the synchronous sensing of both mass perturbation and driving amplitude are achievable with a mass sensitivity of −21.5 Hz/g in the range of 0 mg to 314 mg and a force sensitivity of 8.84 Hz/V in the range of 0.5 V to 2.0 V. Compared to the double amplification mass sensing one proposed in the companion paper, the output voltage, the mass sensing range, and the anti-driving fluctuation performance of the proposed scheme in this paper, are improved or amplified by about 167%, 388%, and 3470% times respectively. It is thus more universal and applicable to various micro and nano resonators, when the Duffing nonlinearity is practically considered.

Experimental and numerical investigation of energy harvesting from double cantilever beams with internal resonance

Conventional harvester systems usually have linear behavior. One of the great matters in these systems is that energy is harvested in a low frequency bandwidth. In this study, double cantilever beams connected at the end through a linear spring are used for energy harvesting. The bimorph piezoelectric patches are attached to the top beam by using double-sided tapes for energy harvesting. A pair of magnets is utilized for adding a nonlinear effect to the system. One of the magnets is mounted at the end of the bottom beam, and the other one is placed on the system’s base. The whole system is designed based on the internal resonance phenomenon. The system’s vibration equations are derived by using the assumed modes method and solved with the numerical solution. Two cases are investigated, including the absence and the presence of the magnetic force. When the magnetic force exists, the effects of nonlinearity and internal resonance on the energy harvesting are investigated and the results are compared to the case when the magnetic force does not exist, which has a linear behavior. To validate the results, a test rig has been designed and fabricated for experimental studies. Experimental results show that the simulation results are in good agreement with the experimental data. The obtained results show that energy is usually harvested in a broader frequency bandwidth when the magnetic force exists compared to when it does not exist. However, when the magnetic force exists, peak voltage can be higher or lower than the case that the magnetic force does not exist. It depends on the system’s parameters.

Research of the internal resonances on a nonlinear dual-rotor based on the energy tracks shifting

The complex coupling characteristics of a dual-rotor system make the mechanism of nonlinear vibrations different from that of a single rotor system. Research of nonlinear vibration of the dual-rotor is beneficial to improve the stability of an aero-engine. Based on nonlinear spring characteristics of rotors, a novel nonlinear dual-rotor dynamics model coupled by the inter-shaft bearing is proposed, and internal resonances on the dual-rotor system are systematically studied. Considering nonlinearities, internal resonance phenomena occurring in the vicinity of the coupling critical rotational speed are systematically analyzed by numerical simulation results such as time histories, Poincare maps, bifurcation maps and largest Lyapunov exponents. Theoretical solutions are calculated based on the improved shooting method and stabilities of theoretical solutions are investigated by using the Floquet theory. The theoretical solutions are consistent with numerical simulation results well. The concepts of the energy track shifting, the energy-Poincare map, stabilities of energy tracks and the energy supplying function are introduced to analyze nonlinear vibration characteristics for the first time. The internal resonance phenomena of the low pressure rotor and the complex nonlinear vibrations of the high pressure rotor are qualitatively analyzed based on the energy viewpoint, and the energy tracks are verified by experiments. Research shows that energy tracks can characterize vibration characteristics well and the energy supplying function provide a possible way to quantify the energy transfer between rotors in the multi-rotor system.

Internal signal stochastic resonance of a two-component gene regulatory network under Lévy noise

Noises are ubiquitous in nature and can often induce some curious phenomena. In this paper, we investigate the internal signal stochastic resonance (ISSR) phenomenon of a two-component gene regulatory network under the excitation of Levy noise. Our results reveal that the Levy noise can induce the periodic oscillation of the protein concentration when the control parameter is close to its bifurcation points. Furthermore, we consider the noise-induced periodic signal as the periodic excitation and study the ISSR phenomenon under the cooperation of the nonlinear system, noise-induced periodic signal and random noise. And we found that there may be a connection between the ISSR phenomenon and the bifurcation mechanism. Besides, we also investigate the effects of different noise parameters on the ISSR phenomenon. The simulation results indicate that there is an optimal interval of the stability index $$\alpha $$ which can induce the ISSR phenomenon, and the skewness parameter $$\beta $$ has a negative correlation with the ISSR phenomenon. Our results may provide a pathway to uncover the positive functional mechanism of noises in complex gene regulatory networks.

Propagation of Overvoltages in the Form of Impulse, Chopped and Oscillating Waveforms in Transformer Windings—Time and Frequency Domain Approach

This paper describes a comparison of overvoltage propagation in transformer windings. Expanding and evolving electrical networks comprise various classes of transient waveforms, related to network reconfigurations, failure stages and switching phenomena, including new sources based on power electronics devices. In particular, the integration of renewable energy sources—mainly solar and wind—as well as expanding charging and energy storage infrastructure for electric cars in smart cities results in network flexibility manifested by switching phenomena and transients propagation, both impulse and oscillating. Those external transients, having a magnitude below the applied protection level may have still a considerable effect on winding electrical insulation in transformers, mainly due to internal resonance phenomena, which have been the root cause of many transformer failures. Such cases might occur if the frequency content of the incoming waveform matches the resonance zones of the winding frequency characteristic. Due to this coincidence, the measurements were performed both in time and frequency domain, applying various classes of transients, representing impulse, chopped (time to chopping from 1 µs to 50 µs) and oscillating overvoltages. An additional novelty was a superposition of a full lighting impulse with an oscillating component in the form of a modulated wavelet. The comparison of propagation of those waveforms along the winding length as well as a transfer case between high and low voltage windings were analyzed. The presented mapping of overvoltage prone zones along the winding length can contribute to transformer design optimization, development of novel diagnostic methodology, improved protection concepts and the proper design of modern networks.

Bifurcation phenomenon and multi-stable behavior in vortex-induced vibration of top tension riser in shear flow

Top tension riser is one of the most frequently used and vulnerable equipment in deep-sea petroleum engineering. With experiments and CFD simulations, researchers have found some hysteresis phenomenon and bistable behavior in vortex-induced vibration of the cylinders. However, there is still insufficient research on the bifurcation phenomenon and the multi-stable behavior of the riser vortex-induced vibration. In this study, based on the Van der Pol wake oscillator model, the vortex-induced vibration cross-flow governing equations of the riser fluid structure–coupled system in linear shear flow are established. Then the simplified multi-degrees-of-freedom model with 15-order modes is derived by the Galerkin method. Poincare map is utilized to obtain bifurcation diagram and identify the dynamic property of the top tension riser in a wide flow velocity range. Nonlinear dynamic behavior such as bistable behavior and even tristable behavior are discovered from the bifurcation diagram. Combined with the riser coupled system’s eigen analysis and the three-dimensional spectrum contour, it is found that there are various phenomena of mode interaction in the riser system, including 1:3 internal resonance phenomenon. Such study can provide reference and guide the design and optimization of riser structural parameters.

Design, analysis and experimental study of a T-shaped piezoelectric energy harvester with internal resonance

In this paper, a T-shaped piezoelectric energy harvester (TPEH) is proposed by taking advantages of internal resonance and multimodal techniques. The developed TPEH is composed of a clamped-sliding main beam and a branched vertical cantilever beam at the center. The sliding end of the clamped-sliding beam could move freely along the axial direction, and two piezoelectric patches are attached at both sides of the substrate near the sliding end. An electromechanical coupling model was established based on the extended Hamilton’s method. The simulation results are in good agreement with the experimental results. The simulation and experimental results indicate that the 1:3 internal resonance phenomenon of the TPEH occurs when the excitation frequencies are close to the first resonance of the TPEH. As a result, the energy conversion efficiency can be significantly increased through the high-frequency response at the low-frequency excitation. Meanwhile, the operational bandwidth is widened with the hardening nonlinear response of the proposed TPEH within the first resonance bandwidth. Finally, it was demonstrated that the proposed TPEH is able to harvest practical vibration energy with low-amplitude and low frequency.

Internal Resonance Phenomena in Coupled Ductile Cantilevers With Triple Frequency Ratio–Part I: Experimental Observations

This paper, the first of two companion papers, mainly reports the experimental observations of the internal resonance in a coupled ductile cantilever system from the viewpoint of mass sensing, mass perturbation effect on coupling tongues, and coupling strength effect on the non-linear vibration. In the coupled ductile cantilever system, two geometrically different Cu-based cantilevers, with a frequency ratio of three to one, are coupled by a coupling overhang. A new flow is proposed to determine the bifurcation points (boundary) of coupling range, which is believed to be more suitable for mass sensing under high energy dissipation circumstance. On one hand, the enlargement of the coupling range is founded to be with the increased exciting force, and the lateral leftward shifting of the coupling tongue is observed with increasing the mass perturbation or lengthening the coupling overhang. On the other hand, the enhancement of the coupling strength, by lengthening the coupling overhang, can make the resonant frequency ratio close to the integer one, decrease the area variation of the coupling tongue with mass perturbation, and improve the linearity of mass sensitivity. This implies that the stability of the coupled system might be improved by enhancing the coupling strength. While a mass sensitivity-amplified scheme and its mechanism will be discussed in Part II.

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.

Normal form analysis of bouncing cycles in isotropic rotor stator contact problems

This work considers analysis of sustained bouncing responses of rotating shafts with nonlinear lateral vibrations due to rotor stator contact. The insight that this is an internal resonance phenomena makes this an ideal system to be studied with the method of normal forms, which assumes that a system may be modelled primarily in terms of just its resonant response components. However, the presence of large non smooth nonlinearities due to impact and rub mean that the method must be extended. This is achieved here by incorporating an alternating frequency/time (AFT) step to capture nonlinear forces. Furthermore, the presence of gyroscopic terms leads to the need to handle complex modal variables, and a rotating coordinate frame must be used to obtain periodic responses. The process results in an elegant formulation that can provide reduced order models of a wide variety of rotor systems, with potentially many nonlinear degrees of freedom. The method is demonstrated by comparing against time simulation of two example rotors, demonstrating high precision on a simple model and approximate precision on a larger model.

A widespread internal resonance phenomenon in functionally graded material plates with longitudinal speed

A widespread internal resonance phenomenon is detected in axially moving functionally graded material (FGM) rectangular plates. The geometrical nonlinearity is taken into account with the consideration of von Kármán nonlinear geometric equations. Using d’Alembert’s principle, governing equation of the transverse motion is derived. The obtained equation is further discretized to ordinary differential equations using the Galerkin technique. The harmonic balance method is adopted to solve the above equations. Additionally, stability analysis of steady-state solutions is presented. Research shows that a one-to-one internal resonance phenomenon widely exists in a large range of constituent volume distribution in moving FGM plates. Moreover, it is found that this internal resonance phenomenon can easily happen even though the FGM plates are under extremely small external excitation or with very large damping.

Characterization of a 3DOF aeroelastic system with freeplay and aerodynamic nonlinearities – Part II: Hilbert–Huang transform

Abstract The Hilbert–Huang Transform is used to analyze the nonlinear aeroelastic response of a 2D 3DOF aeroelastic airfoil system with control surface freeplay under transonic flow conditions. Both static and dynamic aerodynamic conditions, i . e . , for accelerating freestream speed, are considered using a linearized aerodynamic model. The main aim of this paper is to provide an in-depth physical understanding of the observed transition between periodic and aperiodic behavior, and the presence of a stable periodic region well below the domain characterized by stable limit cycles. Physical insights towards the forward and backward abrupt transition between aperiodic/chaotic and periodic behavior types appear to be the result of an internal resonance (IR) phenomenon between linear modes followed by a lock-in between linear and nonlinear modes. More specifically, initially a 2:1 IR between linear modes leads to a shift in the frequency composition and dynamic behavior of the system. A secondary effect of the IR can be observed immediately after the exact point of 2:1 IR such that a nonlinear mode locks into a subharmonic of the linear mode which in-turn drives a finite stable periodic region.

Analysis on two types of internal resonance of a suspended bridge structure with inclined main cables based on its sectional model

Abstract This study investigated the internal resonance phenomenon of a suspended bridge structure with a 6-Degree-of-Freedom sectional model. The primary resonance of the second mode of the system under harmonic excitation is firstly studied. The one-to-two internal resonance between the second and third modes, and one-to-three internal resonance between the second and fourth modes may be induced if the corresponding natural frequency ratios are close to 2.0 and 3.0, respectively. Numerical analysis also shows that the two-to-one internal resonance between the third and second modes can be induced under different scenarios of excitation. The first one happens with the second mode being excited, resulting in two response peaks in the frequency response curves . The second one may occur when the third mode is excited with sufficient large excitation where most of the energy input will be transmitted to the second mode. Hopf bifurcation can also be found in the frequency response curve of the system. Lastly, the three-to-one internal resonance between the fourth and second modes is also found when the second mode is excited. The response of the second mode is slightly reduced with a distinct increase in the response of the fourth mode due to the internal resonance. All these behaviors of this dynamic system indicate the meaningful role played by a variety of internal resonances in the design of mega-scale cable-supported bridge structure.

Normal form analysis of stator rub in rotating machinery

This work considers analysis of sustained impacting cycles of rotating shafts with potentially many disks. The insight that this is an internal resonance phenomena makes this an ideal system to be studied with the method of normal forms. However, the presence of arbitrary non smooth nonlinearities due to impact and rub mean that the method must be extended by incorporating an Alternating Frequency/Time (AFT) step to capture nonlinear forces. The process results in an elegant formulation that can model a very wide variety of rotor systems and is demonstrated by comparing against simulation of a contacting overhung rotor.

Nonlinear vibrations of moving functionally graded plates containing porosities and contacting with liquid: internal resonance

Nonlinear vibrations of longitudinally moving functionally graded material (FGM) plates containing porosities and contacting with liquid are investigated. The FGM plates contain porosities because of technical issues during the FGM preparation. Two types of porosity distribution, namely even and uneven distribution, are taken into account. The liquid is assumed to be incompressible, inviscid and irrotational. The velocity potential along with Bernoulli’s equation is employed to describe the dynamic liquid pressure on the plates. Considering geometric nonlinearity described by the von Karman nonlinear theory, the governing equation of the system is developed by utilizing D’Alembert’s principle. Then, the Galerkin method is employed to reduce the governing equation into a set of ordinary differential equations. An approximately analytical analysis on the nonlinear steady-state response is performed by means of the harmonic balance method; the stability of steady-state response is examined by a perturbation analysis. To have an intuitive understanding of parameter effects, this study retains all dimensional parameters in the analytical results. The possibility of appearance of internal resonance is confirmed by linear analysis of the system. For moving porous FGM plates contacting with liquid, a distinctive internal resonance phenomenon can occur between the first two modes, as we shall see. Different from the common internal resonance, the present internal resonance is very sensitive to external force and can be motivated by quite small force. Results are presented to examine the effects of different parameters on the frequency response of the system.

Internal resonance and low frequency vibration energy harvesting

A nonlinear vibration energy harvester with internal resonance is presented. The proposed harvester consists of two cantilevers, each with a permanent magnet on its tip. One cantilever has a piezoelectric layer at its base. When magnetic force is applied this two degrees-of-freedom nonlinear vibration system shows the internal resonance phenomenon that broadens the frequency bandwidth compared to a linear system. Three coupled partial differential equations are obtained to predict the dynamic behavior of the nonlinear energy harvester. The perturbation method of multiple scales is used to solve equations. Results from experiments done at different vibration levels with varying distances between the magnets validate the mathematical model. Experiments and simulations show the design outperforms the linear system by doubling the frequency bandwidth. Output voltage for frequency response is studied for different system parameters. The optimal load resistance is obtained for the maximum power in the internal resonance case. The results demonstrate that a design combining internal resonance and magnetic nonlinearity improves the efficiency of energy harvesting.

Modal active control of Chinese gongs.

Instruments that belong to the gong family exhibit nonlinear dynamics at large amplitudes of vibration. In the specific case of the xiaoluo gong, this nonlinear behavior results in a pitch glide of several modes of the instrument in addition to harmonic distortion and internal resonances. This study applies a linear modal active control to a xiaoluo gong in an attempt to change its sound properties. First, a modal damping control of the fundamental mode based on a linear identification and a state space controller is applied in the small amplitude regime (no pitch glide). Results indicate that modal control influences not only the controlled mode but also the frequency components involved in distortion or internal resonance phenomena. Second, a modal damping control is performed in the large amplitude regime (in the presence of pitch glide). Results show that modal control does not affect the pitch glide. However, the controller becomes effective at a time trigger which is related to the instantaneous frequency.

The Effect of Fishing Basin Construction On the Behaviour of a Footbrdge Over the Port Channel

Abstract The paper analyses possible causes of failure of the rotating footbridge over the Ustka port channel. In July, 2015, strange behaviour of this object was observed in the form of excessive vibrations of bridge platform suspension rods, with the accompanying acoustic effects. A preliminary geotechnical analysis has revealed that this destructive effect was caused by the nearby construction works, namely construction of a fishing basin and communication routes in the area close to the bridge, which affected the bridge lashing rod foundation settings. Ground vibrations generated by certain construction activities were likely to have direct impact on decreasing the bearing capacity of these rods and increasing the susceptibility od the piles to extraction. After detecting the above problems in bridge operation, its geodetic monitoring was started. The data recorded during this monitoring, along with the results of force measurements in the rods, have made the basis for a series of numerical simulations, performed in the Finite Element Method (FEM) formalism. The bridge structure was analysed in the conditions defined as the emergency state. Extreme efforts of bridge elements and its dynamic characteristics were examined. A possible source of strange behaviour of the footbridge during its operation which was recognised during these simulations was the coincidence of the global natural frequency of the entire bridge structure with local vibrations of suspension rods, at the frequency approximately equal to 1 Hz. This situation was likely to lead to the appearance of the so-called internal resonance phenomenon. As a final conclusion of the research, recommendations were formulated on possible object oriented corrective actions.

Internal resonance vibration induced by nonlinearity of primary suspension system in high-speed vehicle system

This paper studies the phenomenon of internal resonance in high-speed vehicle system under high frequency periodic excitations. A numerical model of the vehicle system, taking into consideration the dynamic effects of the primary suspension system and the flexibility of the car-body, is established for the study. An approximate approach incorporating the incremental harmonic balance method with frequency response function is adopted to solve the dynamic responses of the vehicle system in frequency domain. Numerical results show that internal resonance vibration in the vehicle system may occur with certain combinations of design parameters of the vehicle system. The vibration of the car-body and the primary suspension system are significantly amplified with energy transmitted between the natural modes of the car-body and the primary suspension system. Parametric studies on the internal resonance are further explored. Results show that the nonlinearity of the primary suspension spring and the modal damping ratio of the vehicle system play very important roles to the occurrence of internal resonance.