Superharmonic Resonance Research Articles

Model and experimental verification of a four degrees-of-freedom rotor considering combined eccentricity and electromagnetic effects

In this paper, the combined eccentricity includes the mass eccentricity, static radial eccentricity and angular eccentricity in the rotor of a permanent magnet synchronous motor, and their electromagnetic effects are considered. Approximate analytical expressions of the unbalanced magnetic pull and the additional transverse electromagnetic moment are developed for the combined eccentricity. Moreover, a four degrees-of-freedom rotor system dynamics model is established that describes the influences of these electromagnetic forces and the moments on the rotor with combined eccentricity, when it is not mounted at the midspan of the rotor shaft. The comparisons of the degenerate cases of the four degrees-of-freedom model and Jeffcott rotor model with the radial and angle eccentricities are employed, and the simulation results are in good agreement with one another; further, a critical speed analysis is performed. For experimental validation, the operating mode of the rotor is selected carefully. The rotor of the permanent magnet synchronous motor in an electric vehicle is more compact in the axial direction and hence is considered to be a rigid rotor for which the critical speed is above the operating range of the angular velocity; however, half of the critical speed could be located in this range. Thus, super harmonic resonance occurs, which is discussed based on the combined eccentricity model, and the dynamic response is verified by experiments in which the results agree well in terms of the trajectory shape of the rotor compared with the results of the dynamics model.

Nonlinear Dynamics of Cutting Process considering Higher-Order Deformation of Composite Cutting Tool

In this paper, the nonlinear dynamic analysis of the cutting process of composite cutting tool is performed. The cutting tool is simplified to a nonplanar bending rotating shaft. The higher-order bending deformation, structural damping, and gyroscopic effect of cutting tool are considered. It is assumed that cutting tool is subjected to a regenerative two-dimensional cutting force containing the first and second harmonic components. Based on the Hamilton principle, the motion equation of nonlinear chatter of the cutting system is derived. The nonlinear ordinary differential equations in the generalized coordinates are obtained by Galerkin method. In order to analyze the nonlinear dynamic response of cutting process, the multiscale method is used to derive the analytical approximate solution of the forced response for the cutting system under periodic cutting forces. In the forced response analysis, four cases including primary resonance and superharmonic resonance, i.e., Ω ¯ = ω 1 , Ω ¯ = ω 2 , 2 Ω ¯ = ω 1 , and 2 Ω ¯ = ω 2 , are considered. The influences of ratio of length to diameter, structural damping, cutting force, and ply angle on primary resonance and superharmonic resonance are investigated. The results show that nonlinearity due to higher-order bending deformation significantly affects the dynamic behavior of the milling process and that the effective nonlinearity of the cutting system is of hard type. Multivalued resonance curves and jump phenomenon are presented. The influences of various factors, such as ratio of length to diameter, ply angle, structural damping, cutting force, and rotating speed, are thoroughly discussed.

Theoretical modeling on the combination resonance of size-dependent microbeams

The combination resonance of size-dependent microbeams is investigated. Two harmonic forces act on the microbeam, and combination resonance is observed while the excitation frequencies differ from the resonant frequency. Microbeams with two different sources of nonlinearities including three kinds of boundary conditions, clamped-free (nonlinearity comes from large curvature and nonlinear inertial), clamped-clamped, and hinged-hinged (nonlinearity originates from mid-plane stretching-bending coupling), are taken into consideration to have a deep understanding of this phenomenon. A traveling load acting on the microbeam is presented as a special case of combination resonance. The modal discretization technique is applied to discretize the equations of motion, and then the Lindstedt–Poincare method, a perturbation approach, is employed to solve the resultant equations. The conditions for combination resonance are presented, and frequency-response curves and time histories at the resonance point are obtained for microbeams of each boundary condition. Results reveal that different sources of nonlinearities result in different performances of combination resonance. The free vibration part constitutes a large percentage of the final response. Furthermore, the situation of coexistence of combination resonance and superharmonic (or subharmonic) resonance is determined. The special case demonstrates a higher amplitude than the common combination resonance for all the boundary conditions. Parametric studies are then carried out to discuss the effects of the length scale parameter, excitation force as well as its position, and damping on the performance of the microbeam.

Characterization and interaction of geometric and contact/impact nonlinearities in dynamical systems

In this work, we study how a contact/impact nonlinearity interacts with a geometric cubic nonlinearity in an oscillator system. Specific focus is shown to the effects on bifurcation behavior and secondary resonances (i.e., super- and sub-harmonic resonances). The effects of the individual nonlinearities are first explored for comparison, and then the influences of the combined nonlinearities, varying one parameter at a time, are analyzed and discussed. Nonlinear characterization is then performed on an arbitrary system configuration to study super- and sub-harmonic resonances and grazing contacts or bifurcations. Both the cubic and contact nonlinearities cause a drop in amplitude and shift up in frequency for the primary resonance, and they activate high-amplitude subharmonic resonance regions. The nonlinearities seem to never destructively interfere. The contact nonlinearity generally affects the system’s superharmonic resonance behavior more, particularly with regard to the occurrence of grazing contacts and the activation of many bifurcations in the system’s response. The subharmonic resonance behavior is more strongly affected by the cubic nonlinearity and is prone to multistable behavior. Perturbation theory proved useful for determining when the cubic nonlinearity would be dominant compared to the contact nonlinearity. The limiting behaviors of the contact stiffness and freeplay gap size indicate the cubic nonlinearity is dominant overall. It is demonstrated that the presence of contact may result in the activation of several bifurcations. In addition, it is proved that the system’s subharmonic resonance region is prone to multistable dynamical responses having distinct magnitudes.

An effective crack position diagnosis method for the hollow shaft rotor system based on the convolutional neural network and deep metric learning

In recent years, the crack fault is one of the most common faults in the rotor system and it is still a challenge for crack position diagnosis in the hollow shaft rotor system. In this paper, a method based on the Convolutional Neural Network and deep metric learning (CNN-C) is proposed to effectively identify the crack position for a hollow shaft rotor system. Center-loss function is used to enhance the performance of neural network. Main contributions include: Firstly, the dynamic response of the dual-disks hollow shaft rotor system is obtained. The analysis results show that the crack will cause super-harmonic resonance, and the peak value of it is closely related to the position and depth of the crack. In addition, the amplitude near the non-resonant region also has relationship with the crack parameters. Secondly, we proposed an effective crack position diagnosis method which has the highest 99.04% recognition accuracy compared with other algorithms. Then, the influence of penalty factor on CNN-C performance is analyzed, which shows that too high penalty factor will lead to the decline of the neural network performance. Finally, the feature vectors are visualized via t-distributed Stochastic Neighbor Embedding (t-SNE). Naive Bayes classifier (NB) and K-Nearest Neighbor algorithm (KNN) are used to verify the validity of the feature vectors extracted by CNN-C. The results show that NB and KNN have more regular decision boundaries and higher recognition accuracy on the feature vectors data set extracted by CNN-C, indicating that the feature vectors extracted by CNN-C have great intra-class compactness and inter-class separability.

Investigation on the Nonlinear Vibration Characteristics of Current-Carrying Crescent Iced Conductors under Aerodynamic Forces, Ampere’s Forces, and Forced Excitation Conditions

Aiming at the problem of nonlinear vibration of current-carrying iced conductors, the aerodynamic forces are introduced into the previous vibration equation of current-carrying conductors that only considered Ampere’s forces. At the same time, on this basis, a forced excitation load is further introduced to study the influence of dynamic wind on the nonlinear vibration characteristics of current-carrying iced conductors, and a new current-carrying iced conductors system under the combined action of Ampere’s forces, forced excitation, and aerodynamic forces has been established, and the improved theoretical modeling of current-carrying iced transmission lines made the model more in line with practical engineering. Firstly, the model of current-carrying iced conductors was established, and then the vibration equation of the model was derived. And the vibration equation was transformed into a finite dimensional ordinary differential equation by using the Galerkin method. The amplitude-frequency response functions of the nonlinear forced primary resonances and super-harmonic and subharmonic resonances of the system are derived by using the multiscale method. Through numerical calculation, the influence of current-carrying, spacing, wind velocity, tension, and excitation amplitude on the response amplitude when the primary resonance of the system appears is analyzed, and the difference between the two working conditions (considering the aerodynamic forces and without considering aerodynamic forces) is compared. The influence of the variation of current-carrying i on the response amplitude of super-harmonic and subharmonic resonances and the stability of the steady-state solution of forced primary resonance was analyzed. The results show that the response amplitude and the nonlinearilty of system under the action of aerodynamic forces are smaller and weaker than without the action of aerodynamic forces; the variation of line parameters has a certain influence on the response amplitude of conductor and the nonlinearity of system; the response amplitudes of the primary resonance, super-harmonic resonance, and subharmonic resonance increase with the increase in the excitation amplitudes, and the resonance peak is offset towards the negative value of the tuning parameter σ, showing the characteristics of soft spring, and the response amplitudes are accompanied by complex nonlinear dynamic behaviors such as the multivalue and jump phenomenon. The change of current-carrying i has an obvious effect on the nonlinearity of the system. The nonlinear and response amplitudes of the system are also enhanced with the increase in wind velocity. The stability of the system is judged when the primary resonance occurs, and it is found that the response amplitude shows synchronization and the out-of-step phenomenon with the change of tuning parameters. The research results obtained in this paper would help to further improve the theoretical modeling about current-carrying iced lines, and the research of line parameters can give a certain reference value to practical engineering, and it will have a positive effect on the safe operation of high-voltage transmission lines.

Modal Interaction-Induced Parametric Resonance of Stayed Cable: A Combined Theoretical and Experimental Investigation

The cable-stayed bridge is widely used due to its strong spanning capacity and navigability. However, flexible cables parametrically resonated by external excitation may result in instability or even damage to the bridge. To prevent such undesirable resonance, this paper discusses an in-plane modal interaction-induced parametric resonance of the stayed cable excited by the bridge deck vibration via nonlinear dynamic analysis. Based on the nonlinear distributed model, two modal governing equations of the cable are established via the Galerkin method. A certain working condition, when the external excitation frequency is close to the second-order natural frequency of the stay cable while nearly twice the first-order natural frequency, is theoretically and experimentally investigated. Specifically, the frequency response equations are obtained by the multiscale method, and the stability of solutions is examined through the Routh Hurwitz criterion. Theoretical and experimental results show that bridge deck vibration can induce not only the primary and superharmonic resonance of the cable but also the principal parametric resonance. Parametric resonance-induced bifurcations are also observed in the system. Particularly, the energy exchange from second-order primary resonance to first-order principal parametric resonance is found, which can induce the parametric resonance with the response amplitude one to three times higher than that of the primary resonance. This paper also validates the superiority of the present modal interaction model over the traditional single-mode model in practical engineering applications.

The dynamics of deep water subsea lifting operations in super-harmonic resonance via the harmonic balance method

The dynamics of deep water subsea lifting operations experiencing super-harmonic resonance is analysed in this study. The harmonic balance method is used to solve the non-dimensional equation of motion of the system and the results are compared with time domain integration and with an equivalent energy dissipation model for typical subsea lifting scenarios. It is demonstrated that 1:3 and 1:5 super-harmonic resonances represent significant features of the response of the system and can lead to large dynamic forces in the cable, which may violate the structural limits of the system in real operations. The harmonic balance method presents results almost as accurate as the time domain integration but up to 25 to 35 times faster, while the equivalent energy dissipation model is not able to represent the super-harmonic resonances. Consequently, taking into account the dynamics introduced by super-harmonic resonances is necessary in the analysis of subsea lifting operations, as it can be the limiting design criterion in certain scenarios, and the harmonic balance method can be used as a fast and accurate method to solve this problem.

VIBRATION DIAGNOSTICS OF FATIFUE DAMAGE IN STRUCTURAL ELEMENTS OF GAS TURBINE ENGINES AT THE REPAIR STAGE

Vibration diagnostics of damage belongs to the class of non-destructive methods, which usually do not take long time. However, the main problem of vibration diagnostics is relatively low sensitivity to the critical damage of fatigue crack type, which arises because of long time accumulation of plastic deformation. To improve the sensitivity and reliability of vibration diagnostics of damage two methods were considered. First method was based on the fact, that a characteristic feature of vibrations of structural elements with fatigue crack is the occurrence of non-linear resonances (sub- and super-harmonic) and significant non-linearity of vibration response at these resonances. Secon one – on the fact, that quite noticeable in certain cases increase of damping characteristic caused by a crack can be observed. Analytical and experimental studies of these methods were carried out as applied to the blades of aircraft gas turbine engines. As a result of the studies, the intensity of change in parameter of super-harmonic resonance and in damping characteristics at different parameters of crack was determined. Besides, the experimental techniques for vibration testing of turbine blades were developed. There was demonstrated, that the sensitivity of both considered methods of vibration diagnostics is several orders of magnitude higher than the sensitivity of conventional methods based on the change in natural frequencies and mode shapes, and they can be effectively used for the diagnostics of blades on the stage of engine repair. All the other conditions being equal, second-order superharmonic resonance demonstrates a higher sensitivity to the presence of cracks than the damping characteristic. The smallest crack with an area of 0.2% considered in the work causes a reliably recorded non-linearity of vibration response at superharmonic resonance of order of 2/1. At this the change of damping characteristic slightly exceeds the error of experiment.

Research on natural vibration responses based on asymmetrical dual-rotor model

The asymmetrical dual-rotor can represent special fault cases of an aero-engine, such as cracked rotors or lost blades, and to understand characteristics of natural vibrations of the system are very important on the design or maintenance stages of the aero-engine. The novel asymmetrical dual-rotor model is put forward based on special fault cases of the aero-engine. In the coupled dynamics model, equations of motion and natural vibration are derived by using Lagrange's equation and the harmonic balance algorithm, systematically. Natural vibration frequencies are studied by using equations of natural vibrations of the model, and coupled natural vibration frequencies between rotors are investigated too. Response curves of coupled phenomena of the model are analyzed in detail. Due to effects of an asymmetry and coupling actions between rotors, complex natural vibration phenomena, such as three unstable regions and two super harmonic resonances on the low pressure (LP) rotor, are firstly investigated on the proposed novel dual-rotor model. In addition, solutions of theoretical analyses are good consistent with results of numerical simulations, and results of experiments can further illustrate natural vibration characteristics of the dynamics model. Research results prove the correctness of the proposed coupled dynamics model.

Efficiency of mono-stable piezoelectric Duffing energy harvester in the secondary resonances by averaging method, Part 2: Super-harmonic resonance

In the paper, the potential of harvesting vibratory energy via a mono-stable Duffing energy harvester with base excitation in super-harmonic resonance is investigated. The averaging method is exploited to provide an approximate analytical solution for the system. The displacement and voltage responses are determined depending explicitly on the displacement amplitude and excitation frequency. The explicit approximate analytical expressions of the peak amplitude and corresponding base excitation frequency are also presented. Next, a detailed power flow analysis in super-harmonic resonance is made, then formulations for input mechanical power and energy, useful electrical power and energy, and efficiency are obtained, respectively. Numerical simulations are carried out to show qualitative comparisons between sup- and sub-harmonic cases and linear cases at corresponding base excitation frequencies. It is obtained that super- and sub-harmonic resonances are effective solutions to enlarge the performance of nonlinear piezoelectric energy harvester in wide-band excitation, especially compared to linear one that only works at a single frequency.

Vibrational Amplitude Frequency Characteristics Analysis of a Controlled Nonlinear Meso-Scale Beam

Vibration response and amplitude frequency characteristics of a controlled nonlinear meso-scale beam under periodic loading are studied. A method including a general analytical expression for harmonic balance solution to periodic vibration and an updated cycle iteration algorithm for amplitude frequency relation of periodic response is developed. A vibration equation with the general expression of nonlinear terms for periodic response is derived and a general analytical expression for harmonic balance solution is obtained. An updated cycle iteration procedure is proposed to obtain amplitude frequency relation. Periodic vibration response with various frequencies can be calculated uniformly using the method. The method can take into account the effect of higher harmonic components on vibration response, and it is applicable to various periodic vibration analyses including principal resonance, super-harmonic resonance, and multiple stationary responses. Numerical results demonstrate that the developed method has good convergence and accuracy. The response amplitude should be determined by the periodic solution with multiple harmonic terms instead of only the first harmonic term. The damping effect on response illustrates that vibration responses of the nonlinear meso beam can be reduced by feedback control with certain damping gain. The amplitude frequency characteristics including anti-resonance and resonant response variation have potential application to the vibration control design of nonlinear meso-scale structure systems.

Phase resonance nonlinear modes of mechanical systems

The resonances of forced dynamical systems occur when either the amplitude of the frequency response undergoes a local maximum (amplitude resonance) or phase lag quadrature takes places (phase resonance). This study focuses on the phase resonance of nonlinear mechanical systems subjected to single-point, single-harmonic excitation. In this context, the main contribution of this paper is to develop a computational framework which can predict the mode shapes and oscillation frequencies at phase resonance. The resulting nonlinear modes are termed phase resonance nonlinear modes (PRNMs). A key property of PRNMs is that, besides primary resonances, they can accurately characterize superharmonic, subharmonic and ultra-subharmonic resonances for which, as shall be shown in this paper, phase lags at resonance may be different from π∕2. The proposed developments are demonstrated using one- and two-degree-of-freedom systems featuring a cubic nonlinearity.

Nonlinear free and forced vibration of carbon nanotubes conveying magnetic nanoflow and subjected to a longitudinal magnetic field using stress-driven nonlocal integral model

In this paper primary and secondary resonance of carbon nanotube conveying magnetic nanofluid and subjected to a longitudinal magnetic field resting on viscoelastic foundation with different boundary conditions is investigated. To investigate the small scale effects, stress driven nonlocal integral model has been used and to show the more correctness of stress driven nonlocal integral model response, in studying the behavior of carbon nanotube with different boundary conditions, its results are compared with strain gradient model. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Kármán strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The nonlinear natural frequencies are obtained from the perturbation method and the divergence and flutter instability due to the increase in nanofluid velocity is investigated. Then the frequency response for primary, subharmonic and superharmonic resonance is obtained using the method of multiple scales.Finally, the effects of length small scale parameters, longitudinal magnetic field, magnetic nanofluid and boundary conditions on nonlinear free and forced vibration of carbon nanotube are investigated. As the most important results, as the intensity of the magnetic field increases, the critical flow velocity increases and divergence and flutter occur later. But the critical flow velocity decreases with increasing the intensity of the magnetic field for a carbon nanotube conveying magnetic nanofluid. In forced vibration, increasing the intensity of the magnetic field increases the amplitude of the response for all boundary conditions in primary and secondary resonance.

Semi-numerical analysis of a two-stage series composite planetary transmission considering incremental harmonic balance and multi-scale perturbation methods

Abstract. This study conducts an analytical investigation of the dynamic response characteristics of a two-stage series composite system (TsSCS) with a planetary transmission consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, which solves the closed solution of incremental parameters passing through the singularity point of the analytical path, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method to verify the feasibility and effectiveness of the improved method. Following that, the multi-scale perturbation (MsP) method is applied to the TsSCS proposed in this subject to analyze the parameter excitation and gap nonlinear equations and then to obtain the analytical frequency response functions including the fundamental, subharmonic, and superharmonic resonance responses. The frequency response equations of the primary resonance, subharmonic resonance, and superharmonic resonance are solved to generate the frequency response characteristic curves of the planetary gear system (PGS) in this method. A comparison between the results obtained by the MsP method and the numerical integration method proves that the former is ideal and credible in most regions. Considering the parameters of TsSCS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The effects of the time-varying parameters and the nonlinear deenthing caused by the gear teeth clearance on the amplitude–frequency characteristics of TsSCS components are studied in this special topic.

Acoustic Waves Radiated From Two Degrees-of-Freedom Nonlinear Rigid Oscillator Systems Immersed in Unbounded Compressible Fluid

Abstract This paper is concerned with the nonlinear behaviors of acoustic waves produced by two degrees-of-freedom rigid oscillators containing nonlinearities and immersed in infinite fluid medium. The vibrations of the oscillators are computed by both the harmonic balance method (HBM) and the direct-time integration scheme, whereas the linearized Euler equations (LEEs) of the acoustic fluid are discretized by a fourth-order dispersion-relation-preserving (DRP) scheme in space and a four-level explicit time marching scheme in time. A constrained moving least-squares (CMLS) immersed boundary method (IBM) is used to enforce the boundary conditions on the common interfaces of the rigid oscillators and the Cartesian grid of the acoustic fluid. A serially staggered procedure is adopted to solve the governing equations of the oscillators and the acoustic fluid as a coupled system. The perfectly matched layer (PML) technique is utilized to damp out the out-going acoustic waves on the boundaries of the truncated computational domain to approximate the nonreflecting wave conditions. Physical insights into the mechanism of the nonlinear acoustic waves induced by super-harmonic resonances, principal resonances, internal resonances, and combination resonances of two degrees-of-freedom nonlinear oscillator systems are provided. The interference fringes of the acoustic waves due to the nonlinear vibration of the system are also discussed. Numerical results show that the sound fields radiated from the vibration system with the above nonlinear behaviors exhibit more complicated interference phenomena since the high-order harmonic components are introduced.

Drillstring Oscillations: The Influence of Fluid Loading and Stabilizer Effects

Drillstring vibrations can be undesirable for drilling operations. Here, attention is focused on vibrations of the upper portion of a drillstring as these vibrations can cause drillpipe wear and casing wear. A reduced-order model is developed to study the motions of a drillstring by taking fluid loading and stabilizer effects into account. In this model development, the distributed nature of the fluid loading is taken into account, and the drillstring is treated as a beam structure. Perturbation analyses are carried out with the reduced-order system, and the system responses are examined for primary and secondary (subharmonic and superharmonic) resonance excitations. The analytical-numerical results reveal the rich nature of the system behavior and help understand the drillstring motions during various resonance conditions.

Applying System Dynamics of Discrete Supported Track to Analyze the Rail Corrugation Causation on Curved Urban Railway Tracks

Urban rail corrugation on curved tracks with small radii causes strong howling during operation, which has been bothering subway operating companies for many years. Therefore, revealing its causes and growth is important for the comfort and safety of subway operation. Current studies believe that the occurrence of rail corrugation is largely due to the resonant vibration of the wheel-rail system. However, little attention has been paid to the key causes of the track resonance and the practical prediction of the occurrence probability of rail corrugation on the certain track. This paper intends to solve these above issues. Firstly, the practical model of predicting the rail corrugation growth is proposed based on the wheel-rail coupling interaction, the key causes of corrugation are investigated, and the sensitivity analysis is carried out, while the corrugation superposition model is introduced to the analyze the corrugation evolution as well as to validate the corrugation growth from the aspect of material friction and wear. Secondly, the impact of the key causes on the initiation and development of the rail corrugation is investigated based on the cosimulation. Finally, case studies validate the proposed theory model and method. The results show that the practical prediction model for the rail corrugation growth proposed in this paper is able to estimate the occurrence possibility of rail corrugation on a specific track, and the superharmonic resonance of the track directly excited by passing vehicles eventually leads to rail corrugation. It is also found that shortwave corrugation develops more rapidly, and adjusting the support stiffness or sleeper spacing leads to fluctuations in the corrugation wavelength and its wear rate.

Semi-numerical analysis of a two-stage series composite planetary transmission considering IHB and MsP methods

This study conducts an analytical investigation of the dynamic response characteristics of a two-stage series composite system (TsSCS) with a planetary transmission consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method to verify the feasibility and effectiveness of the improved method. Following that, the multi-scale perturbation (MsP) method is applied to the TsSCS proposed in this paper. The frequency response equations of the primary resonance, subharmonic resonance, and superharmonic resonance are solved to generate the frequency response characteristic curves of the planetary transmission system in this method. A comparison between the results obtained by the MsP method and the numerical integration method proves that the former is ideal and credible in most regions. Considering the parameters of TsSCS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The effects of the time-varying parameters and the nonlinear deenthing caused by the gear teeth clearance on the amplitude-frequency characteristics of TsSCS components are studied in this special topic.

Superharmonic and subharmonic resonances of atomic force microscope subjected to crack failure mode based on the modified couple stress theory

Superharmonic and subharmonic resonances of atomic force microscope subjected to crack failure mode based on the modified couple stress theory