Primary resonance of traveling viscoelastic beam under internal resonance

Under the 3：1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The unde- termined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.

Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method

A homotopy analysis method（HAM）is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic forces.The validity of the HAM is independent of the existence of small parameters in the considered equation.The HAM provides a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter.Two examples are presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincar＇e method and the incremental harmonic balance method.

Primary resonance of fractional-order Duffing–van der Pol oscillator by harmonic balance method

The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.

A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF A PRIMARY RESONANCE OF A THREE CIRCULAR PLATES TORSION VIBRATION SYSTEM

The method of averaging is applied in this paper to deal with primary resonance of a three circular plates torsion vibration system having cubic nonlinearities which is excited by a simple-harmonic excitation. Bifurcation equation of the steady state response is obtained and its singularity analysis is given. The results of theoretical analysis are shown to be in good agreement with experimental ones.

Inter-well internal resonance analysis of rectangular asymmetric cross-ply bistable composite laminated cantilever shell under transverse foundation excitation

The chaotic dynamic snap-through and complex nonlinear vibrations are investigated in a rectangular asymmetric cross-ply bistable composite laminated cantilever shell,in cases of 1:2 inter-well internal resonance and primary resonance.The transverse foundation excitation is applied to the fixed end of the structure,and the other end is in a free state.The first-order approximate multiple scales method is employed to perform the perturbation analysis on the dimensionless two-degree-of-freedom ordinary differential motion control equation.The four-dimensional averaged equations are derived in both polar and rectangular coordinate forms.Deriving from the obtained frequency-amplitude and force-amplitude response curves,a detailed analysis is conducted to examine the impacts of excitation amplitude,damping coefficient,and tuning parameter on the nonlinear internal resonance characteristics of the system.The nonlinear softening characteristic is exhibited in the upper stable-state,while the lower stable-state demonstrates the softening and linearity characteristics.Numerical simulation is carried out using the fourth-order Runge-Kutta method,and a series of nonlinear response curves are plotted.Increasing the excitation amplitude further elucidates the global bifurcation and chaotic dynamic snap-through characteristics of the bistable cantilever shell.

Bifurcation control of nonlinear oscillator in primary and secondary resonance

A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. Bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.

Optimal Delayed Control of Nonlinear Vibration Resonances of Single Degree of Freedom System

The primary resonance of a single-degree-of-freedom(SDOF)system subjected to a harmonic excitation is mitigated by the method of optimal time-delay feedback control.The stable regions of the time delays and feedback gains are obtained from the stable conditions of eigenvalue equation.Attenuation ratio is applied for evaluating the performance of the vibration control by taking aproportion of peak amplitude of primary resonance for the suspension system with or without controllers.Taking the attenuation ratio as the objective function and the stable regions of the time delays and feedback gains as constrains,the optimal feedback gains are determined by using minimum optimal method.Finally,simulation examples are also presented.

Theoretical Explanation for Lower Order Harmonic Resonance Phenomenon of Three-Ring Gear Transmissions

Dynamic investigations revealed that lower order harmonic resonance phenomenon exists in the three ring gear transmission. That is, when the input speed is close to 1/3, 1/6, 1/9,…, 1/3 n of the primary natural frequency of the transmission, the loads on the bearings and gears are especially high. This paper explained this phenomenon from the viewpoint of parametric resonance in terms of perturbation technique. A conclusion was drawn that the basic reason for this phenomenon is the primary resonance caused by forcing excitation and parametric resonance caused by parametric change.

Parametric resonances of convection belt system

Based on the Coriolis acceleration and the Lagrangian strain formula, a gen- eralized equation for the transverse vibration system of convection belts is derived using Newton＇s second law. The method of multiple scales is directly applied to the govern- ing equations, and an approximate solution of the primary parameter resonance of the system is obtained. The detuning parameter, cross-section area, elastic and viscoelastic parameters, and axial moving speed have a significant influences on the amplitudes of steady-state response and their existence boundaries. Some new dynamical phenomena are revealed.

Lagrange-Maxwell equation and magnetic saturation parametric resonance of generator set

Lagrange-Maxwell＇s equation is extended firstly. With the theory of electromechanical analytical dynamics, the magnetic complement energy in air gap of generator is acquired. The torsional vibration differential equations with periodic coefficients of rotor shafting of generator which is in the state of.magnetic saturation are established. It is shown that the magnetic saturation may cause double frequency electromagnetic moment. By means of the averaging method, the first approximate solution and corresponding solution of the primary parametric resonance is obtained. The characteristics and laws of the primary parametric resonance excited by the electromagnetism are analyzed and some of new phenomena are revealed.

Bifurcation analysis for nonlinear multi-degree-of-freedom rotor system with liquid-film lubricated bearings

The oil-film oscillation in dimensional nonlinear problem. In this a large rotating machinery is a complex high- paper, a high pressure rotor of an aero engine with a pair of liquid-film lubricated bearings is modeled as a twenty-two-degree-of-freedom nonlinear system by the Lagrange method. This high-dimensional nonlinear system can be reduced to a two-degree-of-freedom system preserving the oil-film oscillation property by introducing the modified proper orthogonal decomposition （POD） method. The effi- ciency of the method is shown by numerical simulations for both the original and reduced systems. The Chen-Longford （C-L） method is introduced to get the dynamical behaviors of the reduced system that reflect the natural property of the oil-film oscillation.

Forced response of quadratic nonlinear oscillator: comparison of various approaches

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of dif- ferent analytical approximate approaches. The forced vibration of snap-through mecha- nism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincar method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are com- pared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method pre- dicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the Lindstedt- Poincar method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.

Bifurcation of multi-freedom gear system with spalling defect

This study focuses on the bifurcation characteristics of the four degree-of- freedom gear system with local spalling defect to explore the spalling nonlinear dynamic mechanism. The dynamic model of the gear system with spalling defect, time-variant mesh stiffness, and nonlinear clearance is established to investigate the effect of spaUing defect on mesh stiffness and dynamic bifurcation. The primary resonance and internal resonance responses of the spalling model are analyzed by the averaging method, and the bifurcation characteristics with the evolvement of spall and internal excitation are studied by employing the singularity theory for the two-state variable system, which reveal the different bifurcation characteristics caused by the spalling defect. The results obtained herein can provide a theoretical basis to spalling fault diagnosis of gearbox.

Static and dynamic responses of a microcantilever with a T-shaped tip mass to an electrostatic actuation

In this study, nonlinear static and dynamic responses of a microcantilever with a T-shaped tip mass excited by electrostatic actuations are investigated. The electrostatic force is generated by applying an electric voltage between the horizontal part of T-shaped tip mass and an opposite electrode plate. The cantilever microbeam is modeled as an Euler-Bernoulli beam. The T-shaped tip mass is assumed to be a rigid body and the nonlinear effect of electrostatic force is considered. An equation of motion and its associated boundary conditions are derived by the aid of combining the Hamilton principle and Newton’s method. An exact solution is obtained for static deflection and mode shape of vibration around the static position. The differential equation of nonlinear vibration around the static position is discretized using the Galerkin method. The system mode shapes are used as its related comparison functions. The discretized equations are solved by the perturbation theory in the neighborhood of primary and subharmonic resonances. In addition, effects of mass inertia, mass moment of inertia as well as rotation of the T-shaped mass, which were ignored in previous works, are considered in the analysis. It is shown that by increasing the length of the horizontal part of the T-shaped mass, the amount of static deflection increases, natural frequency decreases and nonlinear shift of the resonance frequency increases. It is concluded that attaching an electrode plate with a T-shaped configuration to the end of the cantilever microbeam results in a configuration with larger pull-in voltage and smaller nonlinear shift of the resonance frequency compared to the configuration in which the electrode plate is directly attached to it.

THE ELECTRON SPIN RESONANCE STUDY ON PRIMARY PHOTOCHEMICAL REACTION OF HEMATOPORPHYRIN DERIVATIVE

The rate of oxygen consumption and the yield of free radical anion of hematoporphyrin derivative (HPD) in aqueous solutions of HPD and pyrocatechol were measured by the probe 2,2,6,6-tetramethyl4-piperidone-1-oxyl.It has been found that both singlet oxygen and free radical mechanisms exist simultaneously in primary photochemical reactions, and there is a competition between both mechanisms. When the oxygen concentration in solutions comes down to 12-14% of the stanting level, the predominant mechanism can be changed from the singlet oxygen to the free radical.Whether HPD exists in aggregation state is very important to photosensitization mechanisms.In the presence of the aggregation state of HPD, the predominant mechanism is the free radical,and photosensitization effects of HPD are all the better.

Simulation and experimental research on methods to suppress ferroresonance in potential transformers

Ferroresonance in distribution networks is usually hard to be identified and predicted because it is nonlinear and dependent on multiple factors and conditions.This study discusses different common antiferroresonance methods by numerical simulations and experimental tests.In order to study the reliability of the anti-ferroresonance measures,a 10 kV ferroresonance testing system was setup.Then a number of simulations and experiments were carried out to show the ferroresonance phenomena initiated by single-phase grounding faults and/or single-phase disconnection faults.In addition,potential transformer models with the characteristic of non-linear magnetic excitation were developed and used in PSCAD-based simulation studies.These studies investigated the performance of primary resonance eliminators and arc suppression coils on restraining the ferroresonance under multiple power system parameters and initialisations.In addition,a joint anti-ferroresonance method is proposed and verified.The simulation and experimental studies can provide guidelines for evaluating the anti-ferroresonance methods in distribution networks.

OPTIMAL CONTROL OF NONLINEAR VIBRATION RESONANCES OF SINGLE-WALLED NANOTUBE BEAMS

An optimal time-delay feedback control method is provided to mitigate the primary resonance of a single-walled carbon nanotube （SWCNT） subjected to a Lorentz force excited by a longitudinal magnetic field. The nonlinear governing equations of motion for the SWCNT under longitudinal magnetic field are derived and the modulation equations are obtained by using the method of multiple scales. The regions of the stable feedback gain are worked out by using the stability conditions of eigenvalue equation. Taking the attenuation ratio as the objective function and the stable vibration regions as constrained conditions, the optimal control parameters are worked out by using minimum optimal method. The optimal controllers are designed to control the dynamic behaviors of tile nonlinear vibration systems. It is found that the optimal feedback gain obtained by the optimal method can enhance the control performance of the primary resonance of SWCNT devices.

预应力下具有初始几何缺陷的石墨烯增强多孔复合材料双曲率壳结构的非线性主共振分析

本文研究了石墨烯增强多孔复合材料(GPLRMFs)双曲率壳结构的非线性主共振行为.考虑初始几何缺陷和预应力,并基于Reddy高阶剪切变形壳理论和冯卡门几何非线性,推导出GPLRMFs双曲率壳的非线性运动方程.接着,考虑简支边界条件,利用伽辽金法进行离散得到了非线性常微分方程,随后,利用改进的Lindstedt-Poincare(MLP)法求解,便可以得到GPLRMFs双曲率壳结构的主共振幅频响应关系.在数值分析中,通过与现有文献进行比较,从而验证了本研究的正确性.最后,分析了各个参数(包括壳体类型、石墨烯片(GPLs)分布模式、孔隙率分布类型、初始几何缺陷、GPLs重量分数、孔隙率系数和预应力)对非线性主共振幅频响应曲线的影响.

Vibration characteristics of multi-parametric excitations and multi-frequency external excitations of rolling mill under entry thickness fuctuation of strip

A dynamic rolling force model with multi-parametric excitations and multi-frequency external excitations caused by entry thickness fuctuation of strip was established.Based on the dynamic rolling force,a nonlinear vertical vibration model with multi-parametric excitations and multi-frequency external excitations was established.The method of multiple scales was used to solve the amplitude-frequency characteristic equation of primary resonance of the nonlinear vibration system of a rolling mill.The transition set and the topology structure of systematic global bifurcation were obtained by using the singularity theory.Finally,primary resonance characteristics of the system under entry thickness fuctuation of strip were analyzed by using actual parameters of the rolling mill.The global bifurcation curves with the change of amplitude and frequency of entry thickness fuctuation of strip were obtained by using numerical simulation,and many dynamic behaviors were found such as single-cycle motion,multi-cycle motion and chaotic motion,which can provide a theoretical reference for further restraining the vibration of a rolling mill.

Nonlinear Forced Vibration of Cantilevered Pipes Conveying Fluid

The nonlinear forced vibrations of a cantilevered pipe conveying fluid under base excitations are explored by means of the full nonlinear equation of motion, and the fourth- order Runge-Kutta integration algorithm is used as a numerical tool to solve the discretized equations. The self-excited vibration is briefly discussed first, focusing on the effect of flow velocity on the stability and post-flutter dynamical behavior of the pipe system with parameters close to those in previous experiments. Then, the nonlinear forced vibrations are examined using several concrete examples by means of frequency response diagrams and phase-plane plots. It shows that, at low flow velocity, the resonant amplitude near the first-mode natural frequency is larger than its counterpart near the second-mode natural frequency. The second-mode frequency response curve clearly displays a softening-type behavior with hysteresis phenomenon, while the first-mode frequency response curve almost maintains its neutrality. At moderate flow velocity, interestingly, the first-mode resonance response diminishes and the hysteresis phenomenon of the second-mode response disappears. At high flow velocity beyond the flutter threshold, the frequency response curve would exhibit a quenching-like behavior. When the excitation frequency is increased through the quenching point, the response of the pipe may shift from quasiperiodic to periodic. The results obtained in the present work highlight the dramatic influence of internal fluid flow on the nonlinear forced vibrations of slender Pipes.