Superharmonic Resonance Research Articles

Nonlinear vibration and superharmonic resonance analysis of wind power planetary gear system

Nonlinear vibration and superharmonic resonance analysis of wind power planetary gear system

Sub-harmonic and super-harmonic resonances of spiral stiffened FG cylindrical panels of variable thickness embedded within a nonlinear elastic foundation

This research devotes to study the behaviors of sub- and super-harmonic resonances of spiral stiffened functionally graded (SSFG) variable thickness cylindrical panels (VTCP) exposed to external loading. The panel considered is rested on a nonlinear elastic foundation (NEF), which is composed of two linear foundations namely Pasternak and Winkler foundations, and a nonlinear foundation with cubic stiffness. The cylindrical panels of variable thickness are reinforced with internal spiral stiffeners. Both the cylindrical panels and spiral stiffeners are considered to be continuously graded along their thickness direction. To model the spiral stiffeners, the smeared stiffeners technique is utilized, and for deriving the discretized nonlinear governing equation of the SSFG-VTCP, the improved Donnell shell theory, von-Kármán equation, and Galerkin’s method are applied. In continuing, to examine the sub- and super-harmonic resonances, the method of multiple scales is implemented. Also, to validate the present results, in addition to comparing the present results with that of the previous research, the response of super-harmonic resonance, which is obtained theoretically, is compared with the 4th-order P-T method.

Effect of the Normal Transverse Stress on the Superharmonic Resonance Problem of Three Layered Sandwich Beams with Viscoelastic Core

Effect of the Normal Transverse Stress on the Superharmonic Resonance Problem of Three Layered Sandwich Beams with Viscoelastic Core

Resonance analysis and time-delay feedback controllability for a fractional horizontal nonlinear roller system

<p>In this paper, we investigated the nonlinear vibration characteristics and time-delay feedback controllability of a fractional horizontal roll system, which is described by a fractional Duffing-van der Pol oscillator under an external harmonic excitation. We focused on the resonance of fractional roller systems and conducted corresponding vibration control. The amplitude-frequency equations of primary resonance and superharmonic resonance were obtained using the multiple scale method. The amplitude-frequency characteristic curves of the system with different parameters were presented, and the influence of system parameters on the curves was analyzed. In addition, the time-delay feedback controller was designed to control the parameter excitation vibration. The numerical simulation results have verified the effectiveness of the time-delay controller in eliminating the jumping and hysteresis phenomena of the rolling system. The comparisons of approximate analytical solution and numerical solution was fulfilled, and the result certifies the correctness and satisfactory precision of the approximately analytical solution. The analysis results provide certain theoretical guidance for the vibration reduction of the horizontal nonlinear roller system.</p>

Detection of Closing Cracks in Beams Based on Responses Induced by Harmonic Excitation.

The non-linear contact model was chosen to simulate a closed crack in a cantilever beam. This study examines the shape and characteristics of the phase diagram of a cantilever beam with closed cracks. It investigates how various crack properties influence the geometry of the phase diagram and proposes a method for identifying cracks based on their features. The area of each closed curve in the phase diagram was determined using the pixel method. Based on the results, the contact model proved effective in simulating closed cracks and was sensitive to nonlinear closing cracks. The vibration responses of beams with different damage severities and positions exhibited distinct geometric features. The crack parameter was identified by locating the intersection of contour lines on the maps. According to numerical calculations, the phase diagrams for super-harmonic resonance seem to be more susceptible to changes in closed cracks with varied damage locations and severities. The wavelet transform was also employed to identify closed cracks using RMS signals, and the results were compared with those obtained from the phase diagram.

Rotor crack breathing under unbalanced disturbance

Rotor cracks are a type of failure seriously affecting the safe operation of rotating machinery. It has been reported that cracks under the dominant effect of gravity produce a breathing effect and consequently result in an abnormal vibration of the rotor. However, in practice the rotor cannot avoid unbalance disturbances. As the rotational speed of the rotor continues to rise, unbalance replaces the dominant role of gravity and changes the breathing pattern of cracks. Therefore, this work aimed to study the crack breathing process during the acceleration phase of an unbalanced cracked rotor by coupling the strain energy release rate (SERR) technique with the neutral axis (NA) method. It was found that when the unbalance is dominant, the fracture surface remains fixed instead of being uniformly open and closed. That is, during the entire acceleration phase of the rotor, the crack undergoes a process from uniform breathing to breath-hold breathing. The change of breathing mode induces the whirl phenomenon in the cracked rotor, which gradually disappears with increasing unbalance. This whirl phenomenon can be combined with superharmonic resonance characteristics as an assessment tool for crack fault diagnosis. The above findings are helpful in diagnosing rotor cracking faults under unbalanced disturbances.

Nonlinear analysis and vibro-impact characteristics of a shaft-bearing assembly

This study investigates the nonlinear frequency response of a shaft-bearing assembly with vibro-impacts occurring at the bearing clearances. The formation of nonlinear behavior as system parameters change is examined, along with the effects of asymmetries in the nominal, inherently symmetric system. The primary effect of increasing the forcing magnitude or decreasing the contact gap sizes is the formation of grazing-induced chaotic solution branches occurring over a wide frequency range near each system resonance. The system's nominal setup has very hard contact stiffness and shows no evidence of isolas or superharmonic resonances over the frequency ranges of interest. Moderate contact stiffnesses cause symmetry breaking and introduce superharmonic resonance branches of primary resonances. Even if some primary resonances are not present due to the system's inherent symmetry, their superharmonic resonances still manifest. Branches of quasiperiodic isolas (isolated resonance branches) are also discovered, along with a cloud of isolas near a high-frequency resonance. Parameter asymmetries are found to produce a few significant changes in behavior: asymmetric linear stiffness, contact stiffness, and gap size could affect the behavior of primary resonant frequencies and isolas.

Dynamical behavior of a particle-doped multi-segment dielectric elastomer minimal energy structure

The dynamic behavior of dielectric elastomers (DEs) has significant influence on their performance. The present study investigates the nonlinear dynamics of particle-doped multi-segmented DE minimum energy structures (DEMESs). To simulate the multi-segment DEMES, we consider each segment as a combination of hyperelastic film and elastic beam and obtain the ordinary differential equations governing the system dynamics based on the Euler–Lagrange equations. Due to the difficulty in measuring various physical parameters of DEs in practice, we utilize experimental data from a single-segment DE and employ a physics-informed neural network to predict the unknown parameters of the DE and the framework, such as stiffness K bb and doping volume fraction ϕ. Based on these predictions, nonlinear analysis is performed for the multi-segment system. Stability analyses of the motion equations reveal that the system exhibits a supercritical pitchfork bifurcation with hyperelastic thin film pre-stretching as the bifurcation parameter. For the three-segment DEMES, there are eight stable modes, but only four are illustrated in the bifurcation diagram due to the identical parameter settings for each segment. The amplitude-frequency curves under different AC voltage loads indicate the presence of harmonic, superharmonic, and subharmonic resonances in the system, with varying frequencies and magnitudes depending on the applied load. The Poincaré maps of the time response demonstrate that the system response is predominantly quasiperiodic. Under low voltage loads, the system exhibits periodic oscillations, while under certain high voltage loads, chaotic behavior emerges, characterized by strong nonlinearity in the time-dependent curves and non-periodicity in the Poincaré maps. This study provides insights into the present mathematical model in the motion control of DEMES.

Analysis of vibration characteristics of face gear powering-split transmission system

A nonlinear dynamic model of the face gear powering-split transmission system was established, and the nonlinear dynamic characteristics of the face gear powering-split transmission system were studied by considering various factors such as support stiffness, time-varying meshing stiffness, lateral clearance of teeth, comprehensive transmission error and bearing stiffness. The Runge–Kutta integral method is used to solve the differential dynamics of the system. The nonlinear behavior of the system is described by the phase plane diagram, time history diagram, frequency domain diagram, and Poincare section diagram. The influence of external excitation frequency on the nonlinear behavior of the system is described by the bifurcation diagram. In addition, the multi-scale method is used to analyze the superharmonic resonance of the system and determine the stability conditions of the superharmonic resonance of the system. The influence of meshing damping, time-varying meshing stiffness, and time-delay control parameters on the amplitude-frequency characteristics of the system is studied by numerical analysis of the superharmonic resonance. The results show that the system will show periodic chaotic alternating motion characteristics under external excitation. Selecting meshing damping, meshing stiffness, and time delay control parameters in a reasonable range can effectively reduce the superharmonic co-amplitude and keep the system in a stable state.

Size-dependent nonlinear forced vibration of microbeam using two-phase local/nonlocal viscoelastic integral model with thermal effects

Combination strain-driven (εD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) with classic Kelvin-Voigt viscoelastic model, εD- and σD-two-phase local/nonlocal viscoelastic integral models (TPNVIM) are proposed to study the viscoelastic nonlinear forced vibration behaviors of microbeams in thermal environment. Based on von-Karman large deformation theory and the Hamilton’s principle, the differential governing equations of motion and boundary conditions are derived for nonlinear forced vibration of microbeam. Several non-dimensional variables are introduced to simplify the mathematical formulation. Nonlocal viscoelastic equations in integral form are transferred unitedly into equivalent differential form with constitutive constraints. The axial force of nonlocal microbeam with immovable ends is derived explicitly. Vibration frequency and corresponding linear vibration mode shape are derived through Laplace transformation for linear thermo-elastic vibration of microbeam using εD- and σD-TPNIMs. The Ritz-Galerkin approach is utilized to derive the reduced differential governing equation for nonlinear vibration of microbeam, which has exact same form as single-degree-of-freedom oscillator with cubic nonlinearity when material viscosity is neglected. The method of multiple scales (MMS) based on trigonometric functions is utilized to solve the reduced nonlinear differential equation approximately. The frequency-amplitude responses are addressed for primary and super-harmonic resonances of microbeam. The effects of viscous coefficient, amplitude of excitation force and environmental temperature variation as well as nonlocal length parameter on the nonlinear vibration behavior of microbeam are investigated numerically for different boundary conditions.

The harmonic resonance and singularity analysis of bifurcation for the magnetized elastic plate with action of time-varying magnetic potential

This paper deals with the superharmonic resonance of a ferromagnetic thin rectangular plate in the air-gap magnetic field excited by armature magnetic potential. Electromagnetic forces applied on the plate by the air-gap magnetic field cause the plate to transversal vibrate, which affects the air-gap magnetic field in turn, resulting the vibration and magnetic field coupling. According to the basic electromagnetic field theory and considering the magnetoelastic coupling effect, the air-gap magnetic field intensity is obtained by solving the Laplace's equation satisfied the air-gap magnetic boundary conditions. The electromagnetic force model of soft ferromagnetic plates is determined based on theories of electromagnetic and elasticity. According to the large deflection theory of plates, basic energy relationships and variational equations of the elastic plate are given. Eventually, the nonlinear magnetoelastic vibration equation of ferromagnetic thin plates is derived using Hamilton's principle and Galerkin method. The multi-scale method is used to solve the superharmonic resonance to obtain the amplitude-frequency response equation and the stability discriminant of solutions. The topological analysis of amplitude-frequency equation is carried out using singularity theory, and the bifurcation characteristics of systems on physical parameter planes in different regions are obtained according to the transition set. The correctness of analytical solutions is verified by comparison with numerical solutions. Through numerical calculations, curves of the static deflection and equivalent magnetic force of plate with parameters are given, and the amplitude curves, dynamic phase plane trajectories and time history diagrams of system response with changes in electromagnetic and structural parameters are plotted. Results show that both the decrease of armature magnetic potential amplitude and the increases of plate thickness and initial air-gap thickness reduce the static deflection. The increase of armature magnetic potential amplitude increases the equivalent magnetic force. As the decrease of initial air-gap thickness and the increases of armature magnetic potential amplitude and excitation force amplitude, the amplitudes of the upper branch and lower branch curves representing stable solutions decrease and increase, respectively, and the single-value solution region increases.

Simultaneously primary and super-harmonic resonance of a van der Pol oscillator with fractional-order derivative

The dynamic characteristics and stability of a fractional-order van der Pol oscillator under simultaneously primary and super-harmonic resonance are analyzed by the multiscale method. At first, the first-order approximate analytical solution of the system is obtained, and the analytical results are verified by numerical simulation. In addition, the concepts of equivalent linear damping and equivalent linear stiffness of simultaneously primary and super-harmonic resonance are proposed to enhance the comprehension of fractional parameters roles. Based on Lyapunov's stability method, qualitative analysis is performed on the stability region and the type of equilibrium points of the steady-state solution. The influences of fractional order and coefficient on system stability, equivalent damping, and equivalent stiffness are discussed. It is found that the change of fractional order and coefficient will cause a deviation of the vibration curve and affect the multiple solutions, resonant frequency, and stability range of the system, which is useful to design or control the similar dynamic system.

Harvesting vibration energy by branch structures and amplified inertial force acting on piezoelectric sheets

To improve the efficiency of ambient vibration energy harvesting, a novel vibration energy harvester with branch structures is proposed, which primarily generates electric output through dynamic force rather than deflection. When excited, the branch can amplify the inertial force and acts on piezoelectric sheets perpendicularly. This capability allows the harvester to generate a substantial output without undergoing excessive deflection. Theoretical models are established, and corresponding dynamic equations are derived using the Lagrange equation. The simulation results show that there exists super-harmonic resonance in the response. The validation experiments were carried out, in which the excitation types were chosen as the sweeping-frequency, harmonic and random types sequentially. The experiment results for sweeping-frequency and harmonic excitations demonstrate the occurrence of super-harmonic vibration and resonance in the response. These phenomena lead to a significant increase in output voltages. The experiments involving stochastic excitations demonstrate the harvester's capability to generate large outputs under weak random excitations, with a piezoelectric material with dimensions of 10 × 10 × 0.2 mm3. The root mean square of output power could reach 46.24 μW under the excitation of power spectrum density of 0.06 g2/Hz, about 80 % higher than the classical inverted-beam harvester.

Nonlinear superharmonic resonance and chaotic motion of a moving web under an intermediate nonlinear support

Roll-to-roll manufacturing is the primary means of flexible electronics to facilitate scale-up production, and the web as the printed substrate is guided by the intermediate roller in this process. This paper aims to investigate the nonlinear superharmonic resonance and chaotic motion of a moving web under an intermediate nonlinear support. This support is modeled as a nonlinear elastic spring with linear and nonlinear stiffness, and the equation of motion of the web attached with nonlinear elastic support is derived from the D’Alembert principle and von Karman theory. The resulting equation is reduced to the two-degree ordinary differential equations via the Galerkin truncation, the superharmonic resonance responses of the web system are obtained by the multi-scale method, and the bifurcation analysis and stability are analyzed by the Runge-Kutta numerical method. The results indicate that the linear and nonlinear stiffnesses have a significant effect on amplitude-frequency responses and chaotic motion. This study provides an exploration of vibration behaviors of the web in flexible manufacturing, thereby laying the foundation for the improvement of fabrication productivity.

Superharmonic and triadic resonances in a horizontally oscillated stably stratified square cavity

The response to harmonic horizontal oscillations of a stably stratified fluid-filled two-dimensional square container is examined as the forcing amplitude is increased. For the studied forcing frequency, the response flow at very small forcing amplitudes is a synchronous periodic flow with piecewise-constant vorticity in regions delineated by the characteristics emanating from the corners of the container, regularized by viscosity. The second temporal harmonic of the forced response flow resonantly excites an intrinsic mode of the stratified container, whose magnitude grows as the square of the forcing amplitude. Above a critical forcing amplitude, a sequence of pairs of other container modes are excited via triadic resonances with the second-harmonic-driven mode. The flows are computed from the Navier–Stokes–Boussinesq equations and the ensuing dynamics is analysed using Fourier techniques, providing a comprehensive picture of the transition to internal wave turbulence.

Nonlinear vibration of pinned FGP-GPLRC arches under a transverse harmonic excitation: A theoretical study

The non-linear in-plane primary resonance of functionally graded porous (FGP) sinusoidal arches made of graphene platelet reinforced composites (GPLRC) subjected a transverse harmonic excitation is investigated in this paper. To eliminate the coupling of inner forces and facilitate the derivation, the neutral plane-based equations of motion are established considering the force and bending moment equilibrium conditions. By utilizing an incremental harmonic balance (IHB) technique, the frequency response features, dynamic time history, and limit cycle under primary resonance and 1/2 super-harmonic resonance are determined. FE analysis is conducted to verify the accuracy of the presented results. To further examine the computational efficiency of IHB procedure, a fourth order Runge–Kutta method is used to determine the structural responses as a counterpart. Comprehensive parametric studies are then carried out, particular focus is paid to the cases of material compositions, geometric properties, and dynamic parameters on the frequency response characteristics. It is found that the increasing porosity coefficient significantly exacerbates the left-inclined softening behaviors of the frequency response curves. On the contrary, the left-inclined effect continues to weaken as the GPL weight fraction gradually grows.

Vibration characteristics analysis of flexible helical gear system with multi-tooth spalling fault: Simulation and experimental study

Gear systems operating at high speeds and under heavy loads are susceptible to continuous multiple tooth surface spalling. The time-varying meshing stiffness (TVMS) model of helical gear pair with multi-tooth spalling fault is established by combining image recognition method and loaded tooth contact analysis (LTCA) method to target the actual form of the spalling fault, and its effectiveness was verified through finite element method (FEM). The flexible helical gear dynamic model is established by utilizing 8-node shell model and Timoshenko beam model to simulate the gear foundations and shafts, the TVMS in multi-tooth spalling fault condition is introduced as the excitation. The mapping relationship between the meshing process, TVMS and vibration response under the influence of rotational speed, spalling surface morphology scale, spalling depth and location of spalling occurrence is analyzed by combining simulation and experimental results. As the rotational speed increases, the specific meshing frequency coincides with the natural frequency of the gear system, leading to the phenomenon of superharmonic resonance. Continuous multi-tooth spalling fault results in continuous time-domain shocks and the meshing frequency modulation of the rotational frequency of the gear with spalling fault. In the absence of bottoming out, the increase in spalling surface morphology scale exerts a more pronounced influence on the vibration characteristics of the helical gear system compared to the increase in spalling depth. The research results can provide theoretical basis for gear system fault diagnosis.

Resonance simulation of the coupled nonlinear Mathieu’s equation

Numerous theoretical physics and chemistry problems can be modeled using Mathieu’s equations (MEs). They are crucial to the theory of potential energy in quantum systems, which is equivalent to the Schrödinger equation. According to the mentioned applications, thus, the current study investigates the stability behavior of the nonlinear-coupled MEs. The analysis of the coupled harmonic resonance cases imposes two coupled solvability conditions, which leads to coupled parametric nonlinear Landau equations. In addition, a super-harmonic nonlinear resonance combination is presented. Solutions and stability criteria are discussed for each case. It is shown that resonance produces an unstable system. The transition curves are derived. Numerical calculations show the excitation of the frequency on the periodic solutions.

Effect of negative stiffness nonlinearity on the vibration control effectiveness of tuned negative stiffness inerter damper

Recently, tuned negative stiffness inerter damper (TNSID) integrating the advantages of both negative stiffness and inerter elements has been proposed and verified to be effective for structural vibration control. In previous studies, the stiffness of the negative stiffness element was normally regarded as a constant for simplicity. However, it is well-known that negative stiffness is generally accompanied by nonlinearity, which may lead to undesirable nonlinear vibrations and unstable responses. Therefore, in this paper, the influence of the nonlinearity of negative stiffness, including weakening and strengthening nonlinear negative stiffness, on the performances of two representative TNSIDs with different configurations is investigated. The working mechanisms and optimal parameters of the TNSIDs with simplified linear negative stiffness are introduced firstly. Following that, based on the harmonic balance method, the dynamic responses of the two TNSIDs with different nonlinear negative stiffness are attained. The unstable problems and performance degradation caused by the nonlinearity of negative stiffness are analysed. Parametric studies are further carried out to evaluate the effects of the nonlinear coefficient, excitation amplitude, internal frequency ratio, and damping ratio on the performances of the two nonlinear TNSIDs. Moreover, to eliminate the impact of the negative stiffness nonlinearity, a new structural parameter optimization is conducted by minimizing the maximum displacement amplitude-frequency response of the TNSIDs. The results indicate that the nonlinearity of negative stiffness could lead to the bend and jump of the displacement amplitude-frequency response curves of TNSIDs, as well as superharmonic resonances, especially when the performance of TNSIDs is sensitive to the variation of negative stiffness. Besides, the unstable areas and the maximum displacement amplitude-frequency responses of TNSIDs are dependent on the excitation amplitude and nonlinear coefficient. To avoid the unstable vibration phenomenon and maintain effective vibration control performance, TNSID-P2 is recommended to be equipped with a weakening negative stiffness under both moderate and strong excitations, while TNSID-P1 should be designed with a weakening negative stiffness and work under a moderate excitation due to its performance being highly sensitive to the variation of negative stiffness.

Nonlinear dynamics of an artificial muscle with elastomer–electrode inertia: Modelling and analysis

Artificial muscles are dielectric elastomeric system that mimic the response of the natural muscle and are helpful in many bio-mimicking applications. The existing literature is based on the assumption that electrodes are compliant and are focused on materials with greater extensibility than biological matter. The present work deals with the modelling and nonlinear dynamic analysis of dielectric elastomers considering the elastomer–electrode inertia and damping effect for bio-mimicking. The distinguishing point of the proposed work is the appearance of terms related to the elastomer–electrode inertia effect in the governing equation. Further, the biological muscle’s nonlinear elastic behaviour is incorporated using the Fung material model. Equilibrium stretch variation with parameter exhibits cusp catastrophe. The effect of electrode inertia on the equilibria is examined. Single-well and double-well potential energy characteristics are obtained for constant voltage. The basin of attraction illustrates the sensitiveness of system dynamics on the initial state and damping. The system response, when subjected to time-varying voltage, is explored. Multi-frequency response exists due to harmonic, sub-harmonic, and super-harmonic resonance condition. The resonant frequency variation with geometry and inertia is presented. Time-varying voltage may lead to chaotic behaviour illustrated through the bifurcation diagram. The presence of a strange attractor and positive Lyapunov exponent in the chaotic region further justify the existence of chaos. Additionally, chaotic behaviour is obtained in the case of single-well and double-well potential. Also, the existence of similar attractors with same fractal dimension is shown. The inertia effect significantly changes the dynamic characteristic from chaotic to non-chaotic or vice-versa. The mechanical and electrical load may lead to chaotic or non-chaotic behaviour depending on the geometry of the elastomer. The chaotic region’s dependence on the damping value is also obtained. Our results show that electrode inertia significantly impacts the elastomer under quasi-static and dynamic conditions. The proposed work gives a foundation for precisely designing artificial muscle for practical applications like soft robotics, artificial arms, and different prosthetic implants.