Primary Resonance Research Articles

Magneto-thermoelastic coupled resonance characteristics of a rotating functionally graded ferromagnetic cylindrical shell under double harmonic line loads

The magneto-thermoelastic coupled resonance of a rotating functionally graded (FG) ferromagnetic cylindrical shell subjected to double harmonic line loads in magnetic and temperature fields is investigated. In accordance with Donnell's thin shell theory on the physical neutral surface, the nonlinear geometric equations are determined. Utilizing the thermoelastic constitutive relationships, combining with the gradient characteristics and temperature dependence of physical parameters, the governing equations of kinetic energy and strain energy are given. Based on the magnetoelasticity theory, and considering the nonlinear magnetization of ferromagnetic metal, the calculation formulas of Lorentz force and magnetizing force acting on the shell are deduced. The magneto-thermoelastic coupled dynamic equations are derived by applying Hamilton principle and discretized by Galerkin method. Subsequently, considering the different resonance forms, the multiple scales method and Lyapunov stability theory are employed to obtain the theoretical solutions and stability discriminants. Numerical examples are carried out to plot the characteristic diagrams with different parameters, which are used to analyze the characteristics of nonlinear coupled resonance. Furthermore, the primary resonance and combination resonance are selected to analyze the bifurcation and chaotic behavior. Results show that the variation of different parameters affects obviously the resonance amplitude and stability. The changes of amplitude and position of hard excitation can lead to resonance region drift. Amplitude, frequency, and position of excitation have significant effects on bifurcation and chaos. The motion state of cylindrical shell is transmitted from periodic response to chaotic response through the accumulation of period-doubling bifurcation.

On the influence of an energy harvesting device on a dynamical system

This article studies the motion of a three degrees-of-freedom (DOF) dynamical system, which consists of two parts, a dynamic part, and an electromagnetic harvesting device. The composition of the system depends on the magnet oscillating in the coil of this device. An electric potential is exerted because of the electromotive force to obtain energy harvesting (EH). Lagrange's equations (LE) are used to derive the system’s equations of motion (EOM), which are solved analytically up to a third-order of approximation using the multiple-scales method (MSM). These solutions are considered novel, in which the mentioned method is applied to a novel dynamical system. The gained are confirmed through comparison with the numerical solutions of the EOM to reveal a high degree of consistency. To establish the system’s solvable conditions and modulation equations (ME), three scenarios of primary external resonance are investigated simultaneously. Certain diagrams of time histories for the amplitudes and phases of the examined dynamical system are plotted to show their behaviors at any given instance of time. The Routh–Hurwitz criteria (RHC) are used to establish the stability and instability regions in light of the graphs of resonance response curves. The presence of an energy harvester in dynamic devices helps us to convert vibrational motion into electrical energy that can be exploited in several applications in a variety of daily tasks, including structural and environmental monitoring, military applications, space exploration, and remote medical sensing.

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>

A Novel Bearing Fault Detection by Primary Resonance of Saddle-Node Bifurcation Domains in a Hardening Duffing Oscillator

A Novel Bearing Fault Detection by Primary Resonance of Saddle-Node Bifurcation Domains in a Hardening Duffing Oscillator

A Comprehensive Analysis into the Effects of Quasiperiodicity on the Swing Equation

This research studies the case of quasiperiodicity occurring within the swing equation, a fundamental model that characterises the behaviour of rotor of the machine in synchronous generators in electrical systems. Quasiperiodicity is explained by intricate patterns and understanding the stability of power systems. Bifurcation analysis, frequency domain techniques and numerical simulations are employed to study the swing equation in detail. The objective of this study is to provide a comprehensive understanding of the dynamical behaviour of the equation for the case of quasiperiodicity, using both analytical and numerical methods, when changes are made to the variables of the system. The results show the comparison of primary resonance and quasiperiodicity in the swing equation and analyses the rate at which stability is lost. This will help with the system losing its stability and identifies precursors to chaos which will prevent unavoidable circumstances in the real world.

Nonlinear Vibrations Analysis of Hyperelastic Cylindrical Shells Utilizing the Method of Multiple Scales

In this study, the primary resonance of a hyperelastic thin-walled cylindrical shell subjected to external excitation is examined. The incompressible Moony–Rivlin model, nonlinear Donnell’s shell theory, and Hamilton principle are employed to extract the fundamental equations in three directions. The Galerkin approach is employed to reduce the governing equation into ordinary differential equations form. Then, by neglecting the membrane inertia, three equations are reduced to one equation to analyze the transverse vibrations of the hyperelastic shells. Natural frequencies of different asymmetric modes are used to define the fundamental mode. Then, the multiple scales method is applied to evaluate the frequency–amplitude curves for various parameters. In addition, the model’s accuracy is validated with those presented in the literature for a cylindrical shell and the comparison shows good arrangement results. In the simulation part, the effects of the amplitude of excitation, the ratio of thickness–radius, and the ratio of radius–length on the primary resonances of shells are evaluated. The simulation plots show that for small values of radius-to-length ratios, increasing its value causes to decrease in the stiffness of the thin hyperelastic shell. Also, by increasing the radius–length ratios from 0.3 to 1.18 the hardening of the shell is decreased, and by increasing the radius–length ratios from 1.18 to 2 the hardening behavior is increased.

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.

A 6 degrees of freedom electromagnetic–thermal–dynamic coupling model for high-temperature superconducting levitation system

As the main merits, self-stabilization and no magnetic resistance make the high-temperature superconducting (HTS) magnetic levitation technology a crucial area for high-speed magnetic levitation development. To guarantee a stable operation of superconducting magnetic levitation systems, the dynamic characteristics of superconducting bulk materials occupy a significant place. However, in the previous research, there is still a lack of a simulation method that can describe 6 degree of freedom (DOF) motion of the superconductor. In this paper, an electromagnetic–thermal–dynamic coupling calculation model was established first. Then, the damping characteristics of 5-DOF superconducting levitation were experimentally tested, and the response analysis of the superconductor under 1–20 Hz excitation was carried out to explore the coupled motion relationship between the various degrees of freedom of the superconductor. In addition to the above, the operating conditions and primary resonance intervals that should be avoided by the HTS maglev system were identified. Additionally, a numerical simulation was conducted to investigate the dynamic response of the HTS maglev system under impact loads. All in all, this study explored the temperature rise conditions of superconducting bulk materials under excitation force through magnetic-thermal-force multi-physics coupling research. This 6-DOF model can provide a comprehensive simulation method for superconducting maglev systems in superconductor's motion behavior, attitude, and thermal state monitoring.

Investigation on the 1:2:3 internal resonance of high dimensional nonlinear dynamic system for FGGP reinforced rotating pre-twisted blade under the aerodynamic forces

The study of internal resonances is one of the main research areas in nonlinear dynamics. The energy transfer behaviors between multiple vibrational modes caused by internal resonances lead to complex nonlinear vibration behaviors. In this paper, the rotating blade is modeled as a functionally graded graphene platelet (FGGP) reinforced rotating pre-twisted plate. The nonlinear vibration of a high dimensional nonlinear dynamic system for FGGP reinforced rotating pre-twisted blade under the aerodynamic forces with 1:2:3 internal resonance is investigated. The axial excitations on the blade are primarily caused by axial aerodynamic force in the tip clearance and contact forces at the top of the blade. The subsonic airflow is treated as a transverse excitation and is calculated using the vortex lattice method. Rayleigh-Ritz method is used to obtain the mode shapes of the rotating composite pre-twisted blade. The three-degree-of-freedom ordinary differential equations of the rotating FGGP reinforced pre-twisted plate are derived by using Lagrange equation. Based on the multiscale method, the averaged equations of the rotating FGGP reinforced pre-twisted plate in the case of primary resonance and 1:2:3 internal resonance under axial and transverse excitations are acquired. We present a discussion on the amplitude-frequency response, force-amplitude response, bifurcations, and chaotic motions of a rotating FGGP reinforced pre-twisted cantilever plate under both axial and transverse excitations. A crescent-shaped independent resonance region is found in the amplitude-frequency response curves. The system exhibits complex nonlinear vibration behavior as axial or transverse excitation amplitude changes.

Vibration suppression of pipe conveying fluid using a nonlinear absorber in longitudinal direction

The article outlines the design of a nonlinear absorber that features cubic stiffness. The absorber was created for a pipe conveying fluid and was based on the internal resonance mechanism. To create the nonlinear absorber, a sliding mass oscillator was attached along to the pipe conveying fluid. Nonlinear equations were derived using Hamilton's principle. The flowing fluid was considered both as a constant value and a fluctuating term. To investigate the nonlinear behavior of the coupled pipe-absorber system, the direct multiple scales method was used. By selecting appropriate values of mass and stiffness for the absorber, the natural frequency of the absorber was tuned in three cases: 1–1, 2–1, and 3–1 internal resonance ratios between the natural frequency of the absorber and the first natural frequency of the pipe. The goal of the mechanism was to transfer energy from the pipe to the absorber, and the design was successful in achieving this. A saturation phenomenon was detected in the force modulation response at the three-to-one internal resonance. The results showed a reduction in amplitudes in both frequency and force response curves under the external primary resonance and pulsating flowing fluid in the supercritical regime. The nonlinear absorber has a good performance in a wide range of flow velocity and excitation. The accuracy of the results was verified using Galerkin and Finite difference methods.

Nonlinear vibration of porous truncated conical shell under unified boundary condition and mechanical load

The aim of this article is to research nonlinear vibration responses of a metal porous truncated conical shell undergoing mechanical harmonic loads in thickness direction. The artificial spring technique is employed to unify boundary condition which can arrive at arbitrary classical boundary condition in real-word by changing the stiffness of springs. Utilizing Hamilton's principle, theoretical formulations of the conical shell are developed in view of FSDT and von-Kármán geometric relation. Afterwards, by employing generalized differential quadrature method (GDQM), the governing equation of nonlinear vibration in the form of partial differential equation is truncated and discretized into an ordinary differential equation which is a high-dimensional nonlinear dynamic system described in physical space rather than the mode space. On this basis, the analysis of effect of exciting frequency on the amplitude is carried out for three different distributing types of porous truncated conical shells by numerical integration. The frequency-amplitude responses of the system are depicted utilizing the sweeping frequency. The bifurcation diagrams, time series and phase diagrams are utilized to analyze the nonlinear dynamics of the porous truncated conical shell under unified boundary condition. It is found that the multiple solutions high-incidence area is near the 1st primary resonance, and vibration amplitude is dominated by the 1st order primary resonance and the 3rd one. The hardening-spring nature decreases for Type3, Type2 and Type1 successively.

Decoupling Control of Double-Side Frequency Tuning for LCC/S WPT System

The parameter fluctuations will lead to the frequency detuning problem of the wireless power transfer (WPT) system, thus affecting the transmission efficiency and output stability. The traditional primary-side frequency tuning method cannot realize the full compensation of the secondary-side resonance parameters, and the double-side frequency tuning method with data communication still exists a coupling problem. Therefore, this paper proposes a decoupling control of double-side frequency tuning for LCC/S wireless power transfer (LCC/S WPT) system, which has the advantages of avoiding information exchange and circuit simplicity. Firstly, based on the circuit model, the influence mechanism of frequency detuning on the transmission efficiency and voltage gain are deeply studied. On this basis, a zero phase angle (ZPA) control method based on optimal efficiency tracking and a two-step perturb and observe method with maximum current search are proposed to adjust the primary and secondary resonance parameters respectively. Finally, a 200W experimental prototype was built to verify the effectiveness of the proposed frequency tuning strategy. It is strong robustness to the parameter fluctuations, and double-side can well maintain the resonance state, which can improve the transmission efficiency and stability of the LCC/S WPT system.

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.

Nonlinear Dynamic Analysis of an Electrostatically Actuated Clamped-Clamped Beam and Excited at the Primary and Secondary Resonances.

This work investigates the primary and secondary resonances of an electrostatically excited double-clamped microbeam and its feasibility to be used for sensing applications. The sensor design can be excited directly in the vicinity of the primary and secondary resonances. This excitation mechanism would portray certain nonlinear phenomena and it would certainly lead in increasing the sensitivity of the device. To achieve this, a nonlinear beam model describing transverse deflection based on the Euler-Bernoulli beam theory was utilized. Then, a reduced-order model (ROM) considering all geometric and electrical nonlinearities was derived. Three different techniques involving time domain, fast Fourier transforms (FFTs), and frequency domain (FRCs) were used to examine the appearance of subharmonic resonance of order of one-half under various excitation waveforms. The results show that higher forcing levels and lower damping are required to activate this resonance. We note that as the forcing increases, the size of the instability region grows fast and the size of the unstable region increases rapidly. This, in fact, is an ideal place for designing bifurcation inertia MEMS sensors.

Validity of Galerkin Method at Beam’s Nonlinear Vibrations of the Single Mode with the Initial Curvature

A common strategy for studying the nonlinear vibrations of beams is to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation or equations through the Galerkin method. Then, the oscillations of beams are explored by solving the ordinary differential equation or equations. However, recent studies have shown that this strategy may lead to erroneous results in some cases. The present paper carried out the following three research studies: (1) We performed Galerkin first-order and second-order truncations to discrete the nonlinear partial differential integral equation that describes the vibrations of a Bernoulli-Euler beam with initial curvatures. (2) The approximate analytical solutions of the discretized ordinary differential equations were obtained through the multiple scales method for the primary resonance. (3) We compared the analytical solutions with those of the finite element method. Based on the results obtained by the two methods, we found that the Galerkin method can accurately estimate the dynamic behaviors of beams without initial curvatures. On the contrary, the Galerkin method underestimates the softening effect of the quadratic nonlinear term that is induced by the initial curvature. This may cause erroneous results when the Galerkin method is used to study the dynamic behaviors of beams with the initial curvatures.

Nonlinear Phase Angle and Amplitude Analysis of Porous Functionally Graded Material Rotating Shaft Under Electromagnetic Loads

In this research, the nonlinear response including amplitude and phase angle of a functionally graded inextensional rotating shaft is studied under the electromagnetic load. Three types of porosity distributions through the radial direction are considered in this work. For the rotating shaft with nonlinear curvature and inertia, the governing equations and corresponding boundary conditions are derived. The Galerkin and the multiple scales methods are used to obtain the modulation equations. The effect of the power law index for a functionally graded shaft fabricated from the mixture of ceramic and metal is presented in the frequency response diagrams for primary resonance under electromagnetic force. Numerical simulations include the effects of the power law index of functionally graded material (FGM), porosity distribution, air-gap length, electromagnetic load, and vibration modes of a rotor on the phase angle and amplitude of steady state responses.

Oligodendroglioma of the Brain with a Gliomatous Growth Pattern Mimicking the Radiologic Features of Autoimmune Encephalitis

In this case report of a patient with verified oligodendroglioma with gliomatous growth pattern, at primary clinical and magnetic resonance (MR) diagnosis, the diagnosis of autoimmune encephalitis (AE) was proposed due to the presence of certain radiological patterns typically seen in AE. Further investigation, including advanced MR methods such as magnetic resonance spectrometry, suggested a possible diffuse multifocal glial involvement, which was further confirmed by histological and immunohistochemical examination. The comprehensive approach of evaluating dynamic MRI data, clinical presentation and the use of extended MRI protocols may assist in precise determination of neoplastic changes in the brain tissue.

Magneto-thermo-elastic coupling vibration and bifurcation characteristics of functionally graded rotating cylindrical shell

The magneto-thermo-elastic coupling vibration of rotating ferromagnetic functionally graded cylindrical shell is investigated in this paper. Based on the physical neutral surface theory and Donnell's theory, the expressions of kinetic energy and strain energy are obtained by introducing geometric nonlinearity. The magneto-thermo-elastic vibration equations of the rotating ferromagnetic functionally graded cylindrical shell in multi-physical field are derived via the Hamilton's principle. Considering the magnetization effect of ferromagnetic shell in magnetic field, the expression of magnetization force is obtained according to the nonlinear magnetic constitutive relation. For simply supported boundary conditions, Galerkin's method is used to obtain the discretized ordinary differential equations, and the static deflection of the shell under static load is solved. The steady-state response of the system during primary resonance and the stability of solutions are given by the multi-scale method and the Lyapunov stability theory. Through numerical examples, the characteristic curves and surfaces of amplitude are plotted, and the parameter ranges corresponding to regions of multi-valued solutions and stable solutions are determined. The system dynamics laws describing the coupling effect between multiple parameters such as force, magnetism, heat, and rotational speed are evaluated. The Global bifurcation diagrams and the maximum Lyapunov exponent spectrums under different control parameters are plotted to analyze the effect of parameter changes on the dynamic response of system. The results show that the vibration characteristics of the system are significantly influenced by the external physical fields, in addition to the shell size and material volume fraction. Slight changes in parameters can lead to violent changes in the dynamic response of the system, exhibiting intermittent, jumping, and chaotic characteristics.

Nonlinear dynamics of a piezoelectrically laminated initially curved microbeam resonator exposed to fringing-field electrostatic actuation

In this paper, the nonlinear dynamics of a piezoelectrically sandwiched initially curved microbeam subjected to fringing-field electrostatic actuation is investigated. The governing motion equation is derived by minimizing the Hamiltonian over the time and discretized to a reduced-order model using the Galerkin technique. The modelling accounts for nonlinearities due to the fringing-field electrostatic force, initial curvature and mid-plane stretching. The electrostatic force is numerically computed using finite element simulation. The nonlinear dynamics of the microbeam in the vicinity of primary resonance is investigated, and the bifurcation types are determined by investigating the location of the Floquet exponents and their configuration with respect to the unit circle on the complex plane. The branches on the frequency–response curves, which originate from the period-doubling bifurcation points, are introduced, and the transition from period-1 to period-2 response is demonstrated by slight sweep of the excitation frequency over the time. The effect of DC and AC electrostatic excitation and the piezoelectric excitation on the response of the system are examined, and their effect on the bifurcation types is determined. The force response curves assuming the AC voltage as the bifurcation parameter are also introduced; it is illustrated that in contrast to in-plane electrostatic excitation, in fringing field-based resonators the resonator is not limited by pull-in instability, which is substantially confining the amplitude of the motion in in-plane resonators.

Addressing the Primary and Subharmonic Resonances of the Swing Equation

A research investigation is undertaken to gain a more comprehensive understanding of the primary and subharmonic resonances exhibited by the swing equation. The occurrence of the primary resonance is characterised by amplified oscillatory reactions, voltage instability, and the possibility for system failure. The phenomenon of subharmonic resonance arises when the frequency of disturbance is a whole-number fraction of the natural frequency. This results in the occurrence of low-frequency oscillations and the potential for detrimental effects on equipment. The objective of this study is to expand upon the current literature regarding the impacts of primary resonance and enhance comprehension of subharmonic resonance in relation to the stability of a specific power system model. The analytical and numerical tools are utilised to investigate the fundamental principles of this resonant-related problem, aiming to provide an effective control solution. This choice is driven by the model’s complex nonlinear dynamical behaviour, which offers valuable insights for further analysis. This analysis includes the Floquet Method, the Method of strained parameters, and the concept of tangent instability in order to provide an extension to existing literature relating to primary and subharmonic resonances, taking into account the dynamic and bifurcation characteristics of the swing equation. This objective will be achieved through the utilisation of both analytical and numerical methods, enabling the identification of specific indicators of chaos that can contribute to the safe operation of real-world scenarios.