Nonlinear internal resonance of double-walled nanobeams under parametric excitation by nonlocal continuum theory

In the present work, the nonlinear internal resonance of double-walled nanobeams under the external parametric load is studied. The nonlocal continuum theory is applied to describe the nano scale effects and the nonlinear governing equations are derived by the multiple scale method. The parametric internal resonance is considered and the relation between the frequency and amplitude is discussed. From the numerical simulation, it can be observed that small scale effects are more obvious for short structures. Three different nonlinear cases can be found. The gap between the stable and instable regions is reduced by the van der Walls (vdW) interaction but enhanced by the excitation amplitude. Moreover, the dynamical motions of double-walled nanobeams are sensitive to the initial condition and excitation frequency.     

两自由度非线性振动系统主参数激励下的分岔分析

本文给出了参数激励作用下两自由度非线性振动系统,在1:2内共振条件下主参数激励低阶模态的非线性响应.采用多尺度法得到其振幅和相位的调制方程,分析发现平凡解通过树枝分岔产生耦合模态解,采用Melnikov方法研究全局分岔行为,确定了产生Smale马蹄型混沌的参数值.     

On modelling and analysis of gear drives with nonlinear couplings

The paper deals with modelling of vibration of shaft systems with gears and rolling-element bearings using the modal synthesis method with DOF number reduction. The influence of the nonlinear bearing and gearing contact forces with the possibility of the contact interruption is respected. The gear drive nonlinear vibrations caused by internal excitation generated in gear meshing, accompanied by impact and chaotic motions are studied. The theory is applied to a simple test-gearbox. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Features of the behavior of solutions to a nonlinear dynamical system in the case of two-frequency parametric resonance

Two-frequency parametric resonance in nonlinear dynamical systems is studied by analyzing a delay differential equation with the delay obeying a two-frequency law, which arises in the mathematical simulation of some physical processes. It is shown that the system can exhibit chaotic oscillations (strange attractors) when the parametric excitation frequencies are both close to the doubled eigenfrequency of the system (degenerate case). The formation mechanisms of chaotic attractors are discussed, and the Lyapunov exponents and the Lyapunov dimension are calculated for them. If only one of the parametric excitation frequencies is close to the double eigenfrequency, a two-frequency regime occurs in the system.     

Using delayed damping to minimize transmitted vibrations

In Mokni et al. [Mokni L, Belhaq M, Lakrad F. Effect of fast parametric viscous damping excitation on vibration isolation in sdof systems. Commun Nonlinear Sci Numer Simulat 2011;16:1720-1724], it was shown that in a single degree of freedom system a fast nonlinear parametric damping enhances vibration isolation with respect to the case where the nonlinear damping is time-independent. The present work proposes additional enhancement of vibration isolation using delayed nonlinear damping. Attention is focused on assessing the contribution of a delayed nonlinear damping over a fast parametric damping in terms of minimizing transmissibility. The results show that a nonlinear damping with delay greatly improves vibration isolation.     

时滞影响下受控斜拉索的参数振动稳定性

研究了轴向激励作用下受控斜拉索系统主参数共振的时滞效应,考虑了拉索垂度和几何非线性的影响,基于Hamilton变分原理建立了受控斜拉索系统轴向激励下的非线性参数振动方程,利用Galerkin方法得到时滞动力系统,运用多尺度法对受控系统的主参数共振进行了分析,得到了不同时滞值、控制增益时参数振动稳定域和受控拉索的时程曲线.研究表明,时滞影响下斜拉索振动控制系统的效果变差,参数共振的稳定域发生偏移,对受控斜拉索系统的控制效果随着时滞的增大而变差,从而对控制系统的参数设计起到指导作用.     

Bifurcation and chaos in escape equation model by incremental harmonic balancing

Periodic motions of the nonlinear system representing the escape equation with cosine and sine parametric excitations and external harmonic excitations are obtained by the incremental harmonic balance (IHB) method. The system contains quadratic stiffness terms. The Jacobian matrix and the residue vector for the type of nonlinearity with parametric excitation are explicitly derived. An arc length path following procedure is used in combination with Floquet theory to trace the response diagram and to investigate the stability of the periodic solutions. The system undergoes chaotic motion for increase in the amplitude of the harmonic excitation which is investigated by numerical integration and represented in terms of phase planes, Poincaré sections and Lyapunov exponents. The interpolated cell mapping (ICM) method is used to obtain the initial condition map corresponding to two coexisting period 1 motions. The periodic motions and bifurcation points obtained by the IHB method compare very well with results of numerical integration.     

参外激励下强非线性广义Van der Pol方程的亚谐共振解（英文）

In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.     

Two perturbation formulations of the nonlinear dynamics of a cable excited by a boundary motion

A nonlinear cable excited by an inclined boundary motion, termed as cable's moving boundary problem, is attacked by two different perturbation approaches, i.e., the boundary modulation formulation and the quasi-static drift formulation. The former transforms the boundary motion into a weak modulation on cable's high-order dynamics, while the latter introduces a hybrid mode expansion using an empirical drift shape function. In both formulations, the inclined boundary motion induces three different excitation effects, i.e., longitudinal direct, vertical boundary kinematic, and high-order parametric, all of which being characterized by the parametric modulation factors. Detailed comparative studies indicate that the modulation factors in the two formulations are exactly equivalent to each other only if a new drift shape function, well defined in the boundary modulation formulation, is used for the quasi-static drift formulation. In contrast, the empirical shape functions lead only to an approximate equivalence for intermediate/large boundary motion inclinations. Moreover, for small inclinations, the two formulations induce possible quantitative and qualitative differences. The approximate analytical framework is validated and shown to be computationally efficient, by comparison with the finite difference method.     

Identification of the Nonlinear Restoring Force Characteristic of a Rubber Mounting

This paper is concerned with the identification of the nonlinear restoring force characteristic of a rubber mounting that is part of an oscillator which is subjected to harmonic force excitation. A linearity test is successfully applied to the experimental data to detect the nonlinear behaviour of the system. It is shown that the results of linear system identification depend on the level of excitation force applied. Furthermore, the frequency response functions of the identified linear models only poorly match the measured ones. In consequence, a parametric nonlinear spring characteristic is introduced into the linear model. It is demonstrated that only one set of parameters is required to explain the system behaviour at different levels of excitation force. Also, each of the measured frequency response functions is matched more closely by the calculated frequency response of the nonlinear model.     

Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion

Rotor vibrations caused by large time-varying base motion are of considerable importance as there are a good number of rotors, e.g., the ship and aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle, undergoes large time varying linear and angular displacements as a result of different maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin) generated by different system parameters. Such equations of motion are linear but time-varying in nature, invoking the possibility of parametric instability under certain frequency–amplitude combinations of the base motion. An investigation of active vibration control of an unbalanced rotor–shaft system on moving bases is attempted in this work with electromagnetic control force provided by an actuator consisting of four electromagnetic exciters, placed on the stator in a suitable plane around the rotor–shaft. The actuator does not levitate the rotor or facilitate any bearing action, which is provided by the conventional suspension system. The equations of motion of the rotor–shaft continuum are first written with respect to the non-inertial reference frame (the moving base in this case) including the effect of rotor internal damping. A conventional model for the electromagnetic exciter is used. Numerical simulations performed on the flexible rotor–shaft modelled using beam finite elements shows that the control action is successful in avoiding the parametric instability, postponing the instability due to internal material damping and reducing the rotor response relative to the rigid base significantly, with sufficiently low demand of control current in comparison with the bias current in the actuator coils.     

Chaos of several typical asymmetric systems

The threshold for the onset of chaos in asymmetric nonlinear dynamic systems can be determined using an extended Padé method. In this paper, a double-well asymmetric potential system with damping under external periodic excitation is investigated, as well as an asymmetric triple-well potential system under external and parametric excitation. The integrals of Melnikov functions are established to demonstrate that the motion is chaotic. Threshold values are acquired when homoclinic and heteroclinic bifurcations occur. The results of analytical and numerical integration are compared to verify the effectiveness and feasibility of the analytical method.     

Steady-state response of a geared rotor system with slant cracked shaft and time-varying mesh stiffness

The dynamic behavior of geared rotor system with defects is helpful for the failure diagnosis and state detecting of the system. Extensive efforts have been devoted to study the dynamic behaviors of geared systems with tooth root cracks. When surface cracks (especially for slant cracks) appear on the transmission shaft, the dynamic characteristics of the system have not gained sufficient attentions. Due to the parametric excitations induced by slant crack breathing and time-varying mesh stiffness, the steady-state response of the cracked geared rotor system differs distinctly from that of the uncracked system. Thus, utilizing the direct spectral method (DSM), the forced response spectra of a geared rotor system with slant cracked shaft and time-varying mesh stiffness under transmission error, unbalance force and torsional excitations are, respectively, obtained and discussed in detail. The effects of crack types (straight or slant crack) and crack depth on the forced response spectra of the system without and with torsional excitation are considered in the analysis. In addition, how the frequency response characteristics change after considering the crack is also investigated. It is shown that the torsional excitations have significant influence on the forced response spectra of slant cracked system. Sub-critical resonances are also found in the frequency response curves. The results could be used for shaft crack detection in geared rotor system.     

Effect of fast parametric viscous damping excitation on vibration isolation in sdof systems

In this work, we investigate analytically the effect of cubic nonlinear parametric viscous damping on vibration isolation in sdof systems. Attention is focused on the case of a fast parametric damping excitation. The method of direct partition of motion is used to derive the slow dynamic and steady-state solutions of this slow dynamic are analyzed to study the influence of the fast nonlinear parametric damping on the vibration isolation. This study shows that adding periodic nonlinear damping variation to the vibration isolation device can reduce transmissibility over the whole frequency range. The results also reveals that this nonlinear parametric viscous damping enhances vibration isolation comparing to the case where the cubic nonlinear damping is time-independent.     

Bifurcation and Chaos in a Weakly Nonlinear System Subjected to Combined Parametric and External Excitation

In this paper the dynamics of a weakly nonlinear system subjected to combined parametric and external excitation are discussed. The existence of transversal homoclinic orbits resulting in chaotic dynamics and bifurcation are established by using the averaging method and Melnikov method. Numerical simulations are also provided to demonstrate the theoretical analysis.     

On Stability of a Thin‐Walled Shaft with Active Piezoelectric Fibers

The paper is concerned with the problem of active stabilisation of a rotating flexible shaft made of a composite material containing piezoelectric fibers being controllable by the applied electric field. Rotating shafts exhibit fluttertype instability while exceeding the critical angular velocity. The factor responsible for the loss of stability is internal friction present in the material of the shaft. In the case of a composite structure the internal friction is increased in comparison with steel shafts, and so is the susceptibility of the laminated shaft to self‐excitation. In the paper a method of stabilisation, i.e. shifting the critical threshold towards greater rotation speeds, possibly outside the operating range, is presented. The method is based on incorporation of piezoceramic fibers embedded into the host structure of the shaft. Such integral materials, reflecting the concept of a polymer matrix reinforced with active fibers, are known as Piezoelectric Fiber Composites (PFCs). The carried out examinations have proved that the method is efficient, however limited. It is shown that the critical rotation speed can be increased by several percents, but only within a certain range of structural parameters of the considered system.     

The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration

This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.     

Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness

In this paper, we investigate the Shilnikov type multi-pulse chaotic dynamics for a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time-varying in a periodic form. The dimensionless equation of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions is a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found from the numerical results that there are the phenomena of the Shilnikov type multi-pulse chaotic motions for the rotor-AMB system. A new jumping phenomenon is discovered in the rotor-AMB system with the time-varying stiffness.     

A nonlinear model for rotor-shaft joints of high speed rotor systems with internal damping

Viscous internal damping in joints of high speed rotor systems causes instabilities above a certain frequency of revolution. In the majority of cases a nonlinearity adjusts the stability margin towards higher frequencies. In this paper an analytical solution of a nonlinear four degrees of freedom rotor model with internal damping is proposed, which enables to clearly analyse the influence of shaft stiffness, connection stiffness, rotor mass and shaft mass. The steady state solution of the unbalance case and the stability boundaries are deduced analytically. The stabilizing effect of the nonlinearity is shown. The analytical solutions are in good agreement with numerical results obtained by FERAN, a rotor dynamic simulation tool. A model, representing the rotor–shaft connection with an o–ring has been analyzed by a hydro pulse rig. Beneath the linear way, two further approaches to describe the measured hysteresis, a cubic and a bilinear force law are shown in the paper. The different analytical and numerical results for the whole rotor system with these three approaches are compared. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear vibrations of offshore structures under random wave excitation

An account is given of some mathematical methods which have been used to analyse the response of offshore structures to random wave excitation. The analysis of nonlinear phenomena and the assessment of related nongaussian probability distributions are emphasized. The following problems are analysed in some depth: probability distributions for Morison-type wave loading; response near resonance of nonlinearly damped systems to random excitation; parametric resonance and instability of nonlinearly damped systems with randomly fluctuating restoring force coefficient. Solutions to each of these problem areas are illustrated by practical examples.